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Mass Transport in SingleNanopores
Towards a One-Dimensional State of Neutral Matter
Rene Michel Savard
Department of PhysicsMcGill University
Montreal, Quebec, Canada
December 2014
Under the supervision of Professor Guillaume Gervais
A thesis submitted to McGill University in partial fulfillment of therequirements for the degree of Doctor of Philosophy.
c©2014 Rene Michel Savard
In Knudsen we trust
i
ACKNOWLEDGEMENTS
First, I want to extend my gratitude to some of my colleagues whose contri-
bution is overlooked by the author lists of publications and presentations. I
simply could not have completed my research without them. Jean-Philippe
Masse, from the CM2 at the Ecole Polytechnique de Montreal, has allowed us
to quickly optimize the sample fabrication. He has systematically offered us his
services on very short notice and supported us during long troubleshootings,
and with our out-of-the-ordinary requests with the transmission electron mi-
croscope, focused ion beam and other characterization tools. From the McGill
Nanotools Microfab, the team led by Matthieu Nannini and Don Berry gave
us key insights to help us get out of seemingly inextricable meanders with the
membranes and nanopores. The next person in this list is Pascal Brouseguin,
manager of the McGill Machine Shop, a patient teacher of his craft and an
invaluable colleague to bounce ideas with on all hardware designs. Pascal gave
me concrete proof that rushing into fabrication never pays off.
I want to recognize the support the Department of Physics has provided
to our group through the facilities and staff of the Center for the Physics of
Materials (CPM). To a large extent, this means the help from Richard Talbot,
Robert Gagnon and John Smeros. Without regard to his very busy schedule,
Richard has always been ready to brainstorm and help me come up with the
creative solutions required to make research go forward. Within a moment’s
notice, I could always count on Robert to assist me with new technical pro-
cedures or pick his brains about diverse materials-science techniques. John
was always ready to lend me that extra hand for all the handy-work with our
ii
capricious equipment.
I want to thank James Hedberg for his artistic acumen and the way he
allowed me to learn, through intellectual osmosis and tactful criticism, that
attention to details is crucial when you display your data. Two other “lab
mates” that deserve my thanks are Benjamin Schmidt and Matthieu Bonin,
who were available, day or night, for anything from a quick chat about physics
or, more importantly, for motivational support in harder times.
I would also like to thank the many students working directly at my side.
First off, Charles Tremblay-Darveau, for his support with sorting out the rele-
vant theoretical framework to use in those early days when I barely understood
my experiments. Then Guillaume Dauphinais, for his impressive investigation
of nanopore stability and the countless hours he spent becoming our in-house
expert on nanopore fabrication. I also thank Sam Neale for being a true lab
partner through difficult episodes of our common projects and for his rigorous
work ethics in all aspects of laboratory work. Finally, I thank Pierre-Francois
Duc, for his invaluable help with automating data acquisition and making long
experiments more manageable.
The last, but certainly not least, of the colleagues I want to thank is my
supervisor. Against all odds, Guillaume remained true to his motto: “In
Gervais’ lab, ask for forgiveness, not permission”. The truly collaborative and
intense atmosphere fostered in his group has allowed me to thoroughly enjoy
research, even on late nights and week-ends. As a supervisor, he strove to
foster my critical thought processes when looking at any data and supported
me through my development as a researcher. Guillaume also trusted me with
iii
all aspects of the project allowing me to gain experience in several areas seldom
available to students. This wholistic approach to his supervision is a rare gift
I am very fortunate to have received.
I also acknowledge the financial support from the Natural Sciences and En-
gineering Research Council of Canada (NSERC), from the Fonds de Recherche
du Quebec - Nature et Technologies (FRQNT), and from S. Schulich through
the fellowships in his name.
iv
ABSTRACT
A mathematical description of one-dimensional (1D) many-body systems has
been available for several decades in the form of the Tomonaga-Luttinger Liq-
uids model. The technical means to engineer these 1D systems in the labo-
ratory have, however, only become available recently. Experiments with con-
fined electrons in quantum wires and with anisotropic clouds of cold atoms
have yielded promising results, but limitations on the nature and density of
these particles have prevented some experimental conditions to be attained.
Given this, our purpose here is to develop a new experimental scheme to study
the physics of Luttinger Liquids under conditions outside the reach of other
methods.
In order to achieve this, we investigated the mass flow of helium through
solid-state nanopores. Using a highly-focused electron beam, we fabricated
these nanopores by locally ablating the surface of thin silicon nitride mem-
branes. This tailored approach gave us the ability to produce nanopores as
small as ∼0.5 nm in diameter. We induced flows of helium through these pores
by applying a pressure differential, as large as 50 bar, and we measured the
flow by mass spectroscopy.
The first results I present in this thesis were taken in the gas phase, at
a temperature between 77 and 295 K. In order to demonstrate the feasibility
of the detection of mass flow through single nanopores, we investigated the
conductance of pores in a wide range of diameters, from over 100 nm down to
∼15 nm. We showed how the fluid undergoes a transition from free-molecular
Knudsen effusion to a collective viscous flow as the mean-free-path of the
v
helium decreases. Modeling of the mass flow is shown to be an accurate method
to quantitatively determine the dimensions of nanopores and thus provides an
in situ tool for subsequent experiments at lower temperatures.
In the second half of this thesis, I present our low-temperature results on
the flow of liquid helium in nanopores with diameters of 45 and 16 nm. For
all samples, we detected the onset of superfluidity as a sudden rise in the flow.
Using a two-fluid model of helium, we extracted the superfluid component of
the flow by substracting the contribution of the normal component calculated
with a short-pipe viscous flow equation. This superfluid contribution to the
flow was assumed to be moving at a critical velocity. We found this critical
velocity to be nearly linearly-dependent on temperature and with amplitudes
as large as 20 m/s. These are the largest velocities measured in channel flows
of superfluid, as well as the smallest pores used for strictly direct-current mea-
surements.
Recent quantum Monte Carlo simulations predict the emergence of Lut-
tinger Liquid behavior in solid-state nanopores only a factor of five smaller
than our smallest successful sample. Our experiments therefore represent a
significant step towards a realization of 1D systems of spinless neutral con-
densed matter particles. Our experimental scheme is also ideally positioned to
compare the behaviour of 3He and 4He and thereby investigate the disappear-
ance of bosonic and fermionic quantum statistics in 1D.
vi
ABREGE
Une description mathematique des systemes a plusieurs corps unidimension-
nels (1D) a ete disponible depuis plusieurs decennies sous la forme du modele
des liquides de Tomonaga-Luttinger. Les moyens techniques pour effectuer
l’ingenierie de ces systemes 1D en laboratoire ne sont cependant devenu disponible
que recemment. Des experiences utilisant des electrons confines dans des fils
quantiques, ou encore des nuages anisotropes d’atomes froids, ont produit
des resultats prometteurs, mais certaines limitations sur la nature et la den-
site de ces systemes de particules ont empeche l’atteinte de certain regimes
experimentaux. De ce fait, notre but ici est de developper un nouveau procede
pour l’etude de la physique des liquides de Luttinger dans des conditions hors
d’atteinte des autres methodes.
Pour ce faire, nous avons investigue l’ecoulement massique d’helium a
travers des nanopores fait de matiere solide. En utilisant un faisceau hautement-
focalise d’electrons, nous avons fabrique ces nanopores en effectuant une ab-
lation locale de la surface de membranes minces de nitrure de silicium. Cette
methode nous a donnee la capacite de produire des nanopores aussi petit que
∼0.5 nm de diametre. Nous avons ensuite induit un ecoulement d’helium a
travers ces pores en appliquant un differentiel de pression, jusqu’a 50 bar, et
avons mesure cet ecoulement massique par spectroscopie de masse.
Les premiers resultats que je presente dans cette these ont ete obtenus dans
la phase gazeuse, a une temperature entre 77 et 295 K. De maniere a demontrer
la faisabilite de la detection d’un ecoulement massique a travers un trou unique,
nous avons investigue la conductance de pores de diametres tres varies, de 100
vii
nm a ∼15 nm. Nous avons montre comment le fluide passe par une transi-
tion entre une effusion Knudsen de particules libres a un ecoulement collec-
tif visqueux lorsque la libre-parcours-moyen d’helium rapetisse. Nous avons
demontre que la modelisation de cet ecoulement massique est une methode
precise pour determiner quantitativement les dimensions de nanopores et qu’elle
offre ainsi un outil in situ pour les experiences subsequentes a basses temperatures.
Dans la seconde moitie de cette these, je presente nos resultats a basse
temperature des ecoulements d’helium liquide dans des nanopores ayant un
diametre de 45 et 16 nm. Dans les deux cas, nous avons detecte l’emergence
de la superfluidite sous la forme d’une augmentation soudaine de l’ecoulement.
En utilisant un modele a deux-fluides de l’helium, nous avons extrait la com-
posante superfluide de l’ecoulement en soustrayant la contribution normale cal-
culee avec une equation d’ecoulement visqueux dans des tuyaux-courts. Nous
avons pris pour acquis que cette contribution superfluide de l’ecoulement se
deplacait a une vitesse critique. Nous avons trouve que cette vitesse critique
dependait presque lineairement sur la temperature et que son amplitude pou-
vait etre aussi large que 20 m/s. Ce sont les vitesses les plus larges mesuree
pour des ecoulements superfluides dans des canaux. Ce sont, de plus, les
nanopores les plus petits ayant ete utilises pour des mesures de courant di-
rectes.
Recemment, des simulations Monte Carlo quantiques ont predi l’emergence
de comportements de liquide de Luttinger dans des nanopores de matiere
solide a seulement un facteur cinq des dimensions de nos plus petits pores.
Nos experiences representent donc une etape importante vers la realisation de
viii
systemes 1D pour des particules neutres sans moment angulaire dans la matiere
condensee. Notre procede experimental est, de plus, positionne idealement
pour comparer le comportement des atomes 3He and 4He et ainsi investiguer
la disparition en 1D des statistiques quantiques tels que les bosons et fermions.
ix
STATEMENT OF ORIGINALITY
This thesis contains work that is considered original scholarship and constitutes
a genuine contribution to condensed matter physics. All results previously
published are duly cited and the following list presents the contribution of
authors:
• Apparatus: The author (MS) performed the design of all experimental
cells with the help of Guillaume Gervais (GG). MS and GG designed
the experimental procedure and MS perfomed the implementation of the
measurement apparatus. Filters and other customized components of
the gas handling system were, in large part, machined, designed and
prepared by MS. While all the cooling-specific hardware of the cryostat
was built by Janis Inc., other parts required for mass flow measurement,
including thermometry and heat sinking components, were mounted and
built by the author. The details of the equipment and components are
presented in chapter 4.
• Sample preparation: The silicon nitride membranes were purchased
as ready-to-use substrates and the drilling/imaging of nanopores was ac-
complished either by students from our group or by the TEM technicians.
Specifically, Charles Tremblay-Darveau (CTD) drilled the sample for the
45 nm nanopore used in the Phys. Rev. Lett. 103, 104502 (2009) pub-
lication (chapter 5), and Guillaume Dauphinais (GD) drilled the sample
used in the Phys. Rev. Lett. 107, 254501 (2011) publication (chapter
6). MS handled the large majority of the nanopore preparation after fab-
x
rication, in particular the mounting within the experimental cells. Most
sample fabrication explanations were presented in both publications and
discussed in detail in chapter 4.
• Gas flow measurements: The author participated in all discussions on
the elaboration of experimental protocols, on the interpretation of results
and on the selection of experiments to perform. MS performed most data
acquisition for the gas flow experiments as well as all data analysis for
each sample. GD helped with measurements shown in last section of
chapter 5. The characterization of the 101 nm sample and the modeling
in the transition regime were published, with MS as first author (Phys.
Rev. Lett. 103, 104502 (2009)) and CTD contributed to the development
of the theoretical model. This portion of the thesis represents a first
demonstration of direct mass transport detection in single nanopores
and our samples are the smallest ever studied in continuous channel flow
experiments.
• Superfluid measurement and critical velocities: MS has performed the
data acquisition and analysis for all samples in the superfluid and par-
ticipated to all discussions on the interpretation of results with GG.
Significant contributions to the field of quantum fluid transport include:
- The smallest singular aperture (∼15 nm) allowing a direct detection
of superfluid transport.
- The largest critical velocities observed in channel flow of superfluid
helium.
xi
- The elaboration of an experimental scheme capable of detecting
mass transport in channels small enough to reach the 1D limit.
The superfluid flow results and other characterization of the 45 nm sam-
ple were published as original work, again with MS as first author (Phys.
Rev. Lett. 107, 254501 (2011)).
xii
Contents
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . i
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
STATEMENT OF ORIGINALITY . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction 1
1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Motivation and Historical Context 5
2.1 Fermi Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Tomonaga-Luttinger Liquids . . . . . . . . . . . . . . . . . . . . 7
2.2.1 The Tomonaga-Luttinger Model . . . . . . . . . . . . . . 8
2.2.2 A Class of Luttinger Liquids . . . . . . . . . . . . . . . . 13
2.3 Realizations of 1D Systems . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Electronic Transport in Quantum Wires . . . . . . . . . 15
2.3.2 Laser Traps and Ultra Cold Atoms . . . . . . . . . . . . 17
2.3.3 Helium Adsorbtion in Porous Media . . . . . . . . . . . . 20
2.4 Helium as a Paradigm . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Quantum Monte Carlo Simulations . . . . . . . . . . . . . . . . 24
2.5.1 Comparison to the Luttinger Liquids Model . . . . . . . 25
3 Theory of Mass Flow in Short Pipes 31
3.1 Knudsen Number As an Indicator of Flow Regimes . . . . . . . 32
Contents xiii
3.1.1 Mean-Free-Path of a Gas . . . . . . . . . . . . . . . . . . 32
3.2 A Continuum of Flow Regimes . . . . . . . . . . . . . . . . . . . 33
3.2.1 Knudsen Effusion of Simple Gases . . . . . . . . . . . . . 33
3.2.2 Hydrodynamics of Viscous Flow . . . . . . . . . . . . . . 38
3.3 Liquid Helium Flow . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Excitations in Superfluid Helium . . . . . . . . . . . . . 43
3.3.3 Vortices and Critical Velocity . . . . . . . . . . . . . . . 46
4 Sample Preparation and Flow Measurement Technical Details 51
4.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.1 Solid-State Silicon Nitride Membranes . . . . . . . . . . 52
4.1.2 Sample Fabrication . . . . . . . . . . . . . . . . . . . . . 54
4.2 Flow Measurements Via Mass Spectrometry . . . . . . . . . . . 61
4.3 Measurement Scheme . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 Experimental Cell . . . . . . . . . . . . . . . . . . . . . . 65
4.3.2 Gas Handling System . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Electrical Circuit Analogy . . . . . . . . . . . . . . . . . 71
4.4 Cryogenics and Thermometry . . . . . . . . . . . . . . . . . . . 74
4.4.1 The 3He Cryostat . . . . . . . . . . . . . . . . . . . . . . 75
4.4.2 Thermometry and Thermal Anchoring . . . . . . . . . . 81
5 Flow Conductance of a Single Nanopore 85
5.1 Mass Flow in a Single Nanopore . . . . . . . . . . . . . . . . . . 85
5.1.1 Measurement of Helium Gas Flow . . . . . . . . . . . . . 86
5.1.2 Conductance of a Single 101 nm Nanopore . . . . . . . . 88
5.1.3 Temperature Dependence of Conductance . . . . . . . . 95
5.2 Knudsen Effusion in Smaller Pores . . . . . . . . . . . . . . . . 98
5.2.1 Applicability of the Method in Smaller Nanopores . . . . 98
5.2.2 Evidence of Nanopore Deformation . . . . . . . . . . . . 101
5.2.3 Conductance of the Smallest Single Nanopores . . . . . . 103
5.3 Conclusions from Gas Flow Measurements . . . . . . . . . . . . 106
xiv Contents
6 Hydrodynamics of Superfluid Helium in a Single Nanohole 109
6.1 Knudsen Effusion Below 20 Kelvin . . . . . . . . . . . . . . . . 110
6.2 Liquid Helium Mass Transport . . . . . . . . . . . . . . . . . . . 112
6.2.1 Viscous Fluid Flow . . . . . . . . . . . . . . . . . . . . . 112
6.2.2 Superfluid Flow Through the 45 nm Nanopore . . . . . . 114
6.2.3 Critical Velocity in a Nanopore . . . . . . . . . . . . . . 117
6.3 Liquid Helium Mass Transport in 16 nm Nanopore . . . . . . . 119
6.3.1 Knudsen Effusion Measurement to Confirm Nanopore
Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.2 Superfluid Mass Flow in a 16 nm Nanopore . . . . . . . 123
6.4 Critical Velocities of Superfluids in Single Nanopores . . . . . . 129
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7 Conclusion 133
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Future Work: Towards the Luttinger Regime . . . . . . . . . . . 136
A Appendix: Nanopore Shrinking 139
A.1 Blockages of Nanopores . . . . . . . . . . . . . . . . . . . . . . . 139
A.1.1 Two Types of Nanopore Blockages . . . . . . . . . . . . 140
A.2 Partial Nanopore Shrinking . . . . . . . . . . . . . . . . . . . . 143
A.3 Conditions for Stable Nanopore Fabrication . . . . . . . . . . . 148
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xv
List of Figures
2.1 Occupation number of Fermi levels in 1D and 3D. . . . . . . . . 6
2.2 Dispersion relation of fermions in 1D and their spectra of exci-
tations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Quantum wires engineered within two-dimensional electron gases. 16
2.4 2D optical lattice of cold vapor gases. . . . . . . . . . . . . . . . 18
2.5 Cartoon of the 3D structure of zeolites. . . . . . . . . . . . . . . 20
2.6 P-T phase diagram of bulk helium-4. . . . . . . . . . . . . . . . 23
2.7 Radial density of helium atoms in a nanopore found by quantum
Monte Carlo simulations. . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Density correlations of helium atoms in the core of a nanopore
as a function of the longitudinal position. . . . . . . . . . . . . . 27
2.9 Comparison of quantum Monte Carlo simulations to predictions
of Luttinger liquid theory. . . . . . . . . . . . . . . . . . . . . . 29
3.1 Knudsen scale of flow regimes. . . . . . . . . . . . . . . . . . . . 32
3.2 Decreasing probability of transmission through apertures in the
Knudsen regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Clausing factor for pores with different opening angles. . . . . . 37
3.4 Fraction of normal component in He-II. . . . . . . . . . . . . . . 43
3.5 Superfluid dispersion relation. . . . . . . . . . . . . . . . . . . . 44
3.6 Critical velocity of superfluid helium as a function of channel size. 48
4.1 Photograph of a standard sized TEM wafer. . . . . . . . . . . . 53
xvi List of Figures
4.2 Transmission electron microscope images of single nanopores
fabricated in SiN membranes. . . . . . . . . . . . . . . . . . . . 56
4.3 Inference of the two dimensional profile of a single nanopore
from TEM images. . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Schematic diagram of a mass spectrometer. . . . . . . . . . . . . 63
4.5 3D CAD model of an experimental cell. . . . . . . . . . . . . . . 67
4.6 2D CAD schematic of the inside of the second generation of
experimental cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Photograph of the gas handling system. . . . . . . . . . . . . . . 72
4.8 Cartoon representation of the mass transport measurement sys-
tem and its electrical circuit equivalent. . . . . . . . . . . . . . . 73
4.9 Schematics and photograph of the 3He cryostat insert . . . . . . 78
4.10 Ruthenium oxide thermometer and heat sink for liquid helium . 82
5.1 Volumetric flow as a function of time during decreasing steps of
the applied pressure differential. . . . . . . . . . . . . . . . . . . 87
5.2 Mass flow as a function of pressure applied on a 101 nm diameter
pore. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Conductance of a nanopore with a 101 nm diameter. . . . . . . 90
5.4 Power law exponent of the temperature dependence of conduc-
tance in a 101 nm nanopore. . . . . . . . . . . . . . . . . . . . . 96
5.5 Conductance of nanopores with diameter of 77 nm and 46 nm. . 99
5.6 Conductance of nanopores with a 41 nm and 25 nm diameter. . 102
5.7 Conductance of a nanopore with a 21 nm diameter at 77 K. . . 104
5.8 Conductance of a nanopore with a 15 nm diameter. . . . . . . . 105
6.1 Conductance of a nanopore with a 45 nm diameter as a function
of the Knudsen number. . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Mass flow of liquid helium through a single 45 nm nanopore as
a function of temperature of the experimental cell. . . . . . . . . 113
6.3 Mass flow through a single 15 nm nanopore at 483 mbar of
pressure differential. . . . . . . . . . . . . . . . . . . . . . . . . 115
List of Figures xvii
6.4 Critical velocity of a 45 nm nanopore as a function of tempera-
ture of the superfluid. . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5 Transmission electron microscope images of the 16 nm nanopore
sample used for superfluid flow measurements. . . . . . . . . . . 120
6.6 Gas conductance at 20 K of a 16 nm diameter nanopore inside
the 3He cryostat. . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.7 Superfluid mass flow through a 16 nm sample. . . . . . . . . . . 124
6.8 Superfluid mass flow at 483 mbar in the 16 nm sample. . . . . . 126
6.9 Liquid mass flow through a 16 nm sample as a function of pressure.127
6.10 Critical velocity of the superfluid component in liquid helium
transport through a 16 nm pore. . . . . . . . . . . . . . . . . . . 128
6.11 Critical velocity of superfluid helium as a function of the dimen-
sions of the physical system. . . . . . . . . . . . . . . . . . . . . 130
A.1 Imaging of contaminated samples. . . . . . . . . . . . . . . . . . 141
A.2 Nanopore dimensions as a function of time after fabrication. . . 144
A.3 Diameter of nanopores remaining open as a function of the time
elapsed since fabrication . . . . . . . . . . . . . . . . . . . . . . 146
A.4 Diameter of unstable nanopores as a function of the time elapsed
since fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.5 Phase diagram of the structural stability of nanopores as a func-
tion of the membrane thickness . . . . . . . . . . . . . . . . . . 148
xviii
1
Chapter 1
Introduction
This thesis constitutes the first few chapters of a larger body of work aimed
at unraveling the physics of a one-dimensional (1D) system in the quantum
regime. The ultimate goal is to find a suitable solid state system to confine
test particles in a way for them to experience one-dimensional confinement.
Specifically, any measurement of the properties of a one-dimensional quantum
system is of great importance to determine whether the longstanding theoreti-
cal prescriptions given by Tomonaga-Luttinger liquids theory indeed constitute
a viable model.
In the search for materials where one-dimensional quantum systems could
be realized, promising results have been obtained with electrons propagating
in carefully engineered materials. These heterogeneous crystalline substrates
allow charge carriers to propagate in very thin regions and the application of
additional electric fields can further restrict the areas accessible to the carri-
ers, such that they can be confined to narrow transport channels. Transport
of electrons in sufficiently narrow channels is expected to lead to behavior
2 Introduction
described by the Tomonaga-Luttinger liquids theory [1].
Electrons in a semiconductor structure are however not the only test par-
ticle that can be confined to low dimensions. In fact, recent studies [2, 3]
have shown, using quantum Monte Carlo simulations, that neutral atoms con-
fined in a solid state nanopore can interact such that key parameters of the
Tomonaga-Luttinger theory can be extracted. These computations were in-
spired by the experimental work described here and have predictive power in
the limit where one-dimensional behavior is expected to occur.
The atom of choice for these experiments is helium, in large part because
its large zero-point motion prevents solidification, even at zero Kelvin, thus
allowing experiments with a strongly interacting quantum liquid. Helium is
also available as two isotopes (4He and 3He), which are respectively bosons and
fermions. Helium thus offers the promising possibility to conduct an experi-
ment with either one of the atomic species, and compare behavior of particles
with fundamentally different quantum statistics.
1.1 Outline of the Thesis
The overarching goal of the experiments presented in this thesis is to imple-
ment an experimental scheme where helium atoms would be confined in solid
state nanopores and their physical properties probed by measurement of the
mass transport. Chapter 2 presents the motivations behind this work as well
as a broad overview of the history of theoretical investigations of physical sys-
tems confined to a single dimension. The prevalent experimental attempts to
create a tangible 1D system are also described, as well as a comparison of their
1.1 Outline of the Thesis 3
strengths and weaknesses with respect to our proposed experimental scheme.
The different avenues available to manufacture a highly confined system are
also discussed, along with the rationale behind the choice of the solid state
nanopores used in this thesis. Selected results from extensive computational
studies by our collaborators are also presented in order to offer strong quanti-
tative expectations.
The subsequent chapter is designed to lay down the theoretical background
necessary to complete the analysis of the mass transport in channels. First,
I describe the dynamics of fluid in the different flow regimes as it transitions
from a rarefied effusion to a high density viscous flow. A hydrodynamic model
tailored to the particular geometry of solid-state nanopores is also developed.
Second, I give a simple description of the characteristics of helium at low
temperatures and provide the key knowledge for the interpretation of liquid
and superfluid helium transport.
In chapter 4, I describe the experimental techniques and apparatus re-
quired to implement our measurement scheme. Special attention is given to
the sample fabrication process and how it influences the nanopore geometry.
Indeed, the specific dimensions of the nanopores directly affect the interpreta-
tion of the results obtained. The principles underlying our main measurement
tool, namely a helium mass spectrometer, are then presented alongside a de-
scription of the experimental equipment developed for the sensitive detection
of helium flows. A specific section is also devoted to the cryogenic aspect of
the experiments since working at low temperatures imposes several stringent
constraints on the design of an experimental cell and on the protocol of exper-
4 Introduction
iments. The chapter ends with a description of the thermometry apparatus
and thermalization hardware designed for experiments at low temperatures.
The following two chapters are devoted to the presentation of selected ex-
perimental data and results we obtained throughout our investigations. The
first is composed of results from gaseous flow of helium and serves as a step-
ping stone for the understanding of the liquid flow experiments presented in
the chapter that follows. The gas flow results of chapter 5 first focus on exper-
iments with large nanopores, acting as a proof-of-concept for the experimental
scheme proposed here. Then, a series of experiments that slowly bridge the
gap towards the quasi-1D regime is shown for increasingly small nanopores.
Since the realization of a 1D quantum state requires both low temperatures
and highly confined geometries, the results of chapter 6 are presented in two
parts. First, I show results from experiments that probe the flow of liquid
helium both in the normal and superfluid phase in a 45 nm pore where a fully
3D behavior is expected. Then, in order to approach the 1D limit, I present
results from a nanopore of only ∼ 15 nm across.
The results presented in both chapters 5 and 6 are then summarized and
discussed with respect to future avenues of investigation opened up by this
work. Finally, I also present in the appendix some critical results regarding
the stability of solid-state nanopores. These preliminary findings are relevant
to our efforts to reach the 1D limit.
5
Chapter 2
Motivation and Historical
Context
2.1 Fermi Liquids
The many-body interactions between particles in physical systems are often
the root of exotic phenomena that spark the interest of condensed matter
physicists. Fermi liquid theory is an example of a theory which describes
particles with non-negligible interactions and whose predictions have been rig-
orously tested experimentally. In Fermi liquid theory, the interactions between
fermions cannot be ignored, which makes a simple microscopic description with
independent particles unsuitable. Without those interactions, the many body
system of fermions is simply described by the free-fermion gas model where,
at T=0, the occupation n(k) of all available energy levels is n(k ≤ kF ) = 1 up
to the Fermi level and n(k > kF ) = 0 above. This is depicted in figure 2.1A.
In this non-interacting case, a single fermion can be taken from the Fermi sea
6 Motivation and Historical Context
(k < kF ) and excited into an unoccupied state above the Fermi level. At fi-
nite temperatures, this excitation to a higher energy state is only possible for
particles near the Fermi energy.
Fig. 2.1 (A) Occupation number of the 1D Fermi system of freeelectrons at T=0. (B) Occupation number with interactions be-tween the constituent 1D fermions. (C) Schematic representationof the Fermi sphere in 3D where quasi-particles near the Fermi sur-face can be excited to higher levels. Parts (A) and (B) reproducedfrom [4].
The key insight behind Landau’s Fermi liquid theory is the idea that rele-
vant excitations of a physical system of interacting fermions are also restricted
to particles with an energy near the Fermi level. When this is true, the system
should be well described by a ground state, a filled Fermi sea (figure 2.1A,C)
and a low energy spectrum of excitations near kF . If the interactions are turned
on gently enough (adiabatically), the system’s excitations can be mapped one-
to-one to the original interacting fermions. Importantly, these new mean-field
quasi-particles are non-interacting and the Hamiltonian describing this rede-
fined system is thus more easily solvable. These new quasi-particles are almost
identical to the bare fermions except for a renormalization of some dynami-
cal properties such as, for example, the effective mass. The occupation of
2.2 Tomonaga-Luttinger Liquids 7
states n(k) of a Fermi liquid (figure 2.1B) is thus very similar to that of the
free-fermions model and the properties of the system are again defined by the
quasi-particle excitations near the Fermi level. While Fermi liquid theory is,
strictly speaking, an approximation, it “becomes an asymptotically exact so-
lution for low energies” [5] and its predictions remain very powerful for most
fermionic systems.
One dimensional (1D) quantum systems are, however, not well described
by Fermi liquid theory. In fact, the Fermi liquid model breaks down in 1D
because of a Peierls instability caused by the point-like Fermi surface. A single
wavevector (±2kF ) can connect the two points of the Fermi surface (±kF )
and this produces a divergent response under perturbations, so the mean-field
description of the system is no longer applicable. A different approach is
therefore required to describe the physics of strongly correlated 1D systems.
Fortunately, a treatment of 1D interacting fermions similar to the Fermi liquid
approach was devised decades ago and can be applied to a large class of 1D
systems.
2.2 Tomonaga-Luttinger Liquids
In 1950, a model of interacting fermions in 1D was proposed by Tomonaga
[6]. The model’s assumptions were later redefined by Luttinger [7] and some
subtle mistakes in calculations were eventually corrected by Mattis and Lieb
[8] in the 1960s. The model for spinless and massless interacting fermions
is constructed by using a bosonization of collective excitations to define a
suitable Hamiltonian. Subsequent seminal work by Haldane on spinless 1D
8 Motivation and Historical Context
fermions [9] introduced a generalized model giving insights into the dynamics
of 1D systems in terms of interaction strength between the particles forming a
1D chain. Importantly, Haldane highlighted the possibility of treating several
models of 1D interacting fermions or bosons with the same description of
low energy excitations. The Tomonaga-Luttinger “liquid” was thus named in
reference to the low-energy excitation of particles in the Tomonaga-Luttinger
(TL) model [9]. The process is akin to the emergence of the Fermi liquid model
from a modification of the free-Fermi gas model because of added interactions.
2.2.1 The Tomonaga-Luttinger Model
Fig. 2.2 (A) Dispersion relation of fermions in 1D. Particles areexcited with momentum q and energy E = ω where the allowedexcitation spectra (B) is gapless only for q = 0,±kF . (C) Thesimplified dispersion relation assumed within the Luttinger modelwhere the dotted lines in (A) depict the linear dispersion approx-imation at the region of interest for low-energy excitations exclu-sively. Two types of particles are presumed, right moving and leftmoving and the dispersion relation is assumed to be linear withE = ±vk. Figure based on figures from [5], chapter 3.
The dispersion relation of a 1D fermionic systems is shown schematically in
figure 2.2A, where fermions (full dot) can be excited above the filled Fermi sea,
thus creating a hole (open circle). For low energy, this particle-hole excitation
2.2 Tomonaga-Luttinger Liquids 9
is restricted to fermions near the two Fermi points at ±kF . The excitation
spectrum is shown in fig. 2.2B where we note the important gap for 0 ≤ q ≤
2kF , where the subscript “F” denotes the Fermi level and q is the momentum
transfer during excitation. This feature emerges from the fact that the only
processes that can excite fermions out of the Fermi sea are for |q| = 0, 2kF and
leads to the nesting behavior alluded to in the previous section when discussing
the instability of Fermi liquids in 1D.
The TL model is constructed based on assumptions about the dispersion
relation of the fermions near ±kF . Specifically, the exactly solvable nature of
the TL model relies on the linearized dispersion relation Er(|k|) ' ~(rk−kF )vF
extending down to all lower energies (see figure 2.2C) and the presence of two
types of particles, namely right (r = +1) and left (r = −1) moving ones.
The dotted lines on figure 2.2A represent this linear simplification for the 1D
fermions and figure 2.2C explicitly shows the linear dispersion relation of the
TL model.
The Hamiltonian for this simplified system can be built in two parts [5]:
the first interactionless Hamiltonian H0, and the interaction specific H2 and
H4 such that
HTL = H0 +H2 +H4, (2.1)
and
10 Motivation and Historical Context
H0 = ~∑r,k,s
vF (rk − kF ) : c†rkscrks : (2.2)
H2 =1
L
∑q,s,s′
[g2‖(q)δs,s′ + g2⊥(q)δs,−s′
]ρ+,s(q)ρ−,s′(−q) (2.3)
H4 =1
2L
∑r,q,s,s′
[g4‖(q)δs,s′ + g4⊥(q)δs,−s′
]: ρr,s(q)ρr,s′(−q) : . (2.4)
In the previous Hamiltonian HTL, the fermions described by the operators
c†rks have two possible spin states s and propagate in one of two directions
r = +,− for right and left moving particles respectively. The system has length
L and particles have momentum k up to the Fermi level. The terms within
“: ... :” are normal ordered for fermionic operators. The interactions described
byH2 andH4 are for four-fermions events where a momentum transfer q occurs
and the coupling constants g2(g4) have the usual meaning of forward scattering
between particles of different(same) chirality1. The subscript “‖” refers to
same-spin interaction and the “⊥” one refers to opposite-spins coupling. The
Kronecker delta δ has the usual definition of δ = 1 when indices are the same.
The density fluctuation operators ρr,s =∑
k : c†r,k+q,scr,k,s : define particle
fluctuations with momentum q restricted to 2πm/L for an integer m given a
finite length system.
The present form of the TL Hamiltonian hints at the nature of the excita-
tions in 1D system being of bosonic nature. Given the previous assumptions
about the dispersion of particle-hole pair excitations, the Hamiltonian can
in fact be rewritten as a harmonic oscillator where the eigenstates are truly
1Chirality in this case refers to whether particles are left-moving or right-moving.
2.2 Tomonaga-Luttinger Liquids 11
bosonic. This Bogoliubov transformation of the initial, fermionic states into
operators with bosonic properties is the historical root of the term “bosoniza-
tion”. The details of the latter transformation of the Hamiltonian are presented
in [5, 10].
Looking at the free-fermion case, some important insights can be obtained.
The previous form of H0 in equation 2.1, is equivalent to
H0 =π~vFL
∑r,q 6=0,s
: ρr,s(q)ρr,s(−q) : + constant, (2.5)
where any new excitation in the system is added above the Fermi level and
this excitation adds a significant amount of kinetic energy to this Hamiltonian.
The free-fermion Hamiltonian can be transformed to
H0 =π~vsL
∑r,q 6=0,s
: ρr,s(q)ρr,s(−q) : (2.6)
+π~2L
∑s
[vN(N+,s +N−,s)
2 + vJ(N+,s −N−,s)2],
where Nr,s ≡ ρr,s measures excitations restricted to q = 0 and vs = vN = vJ =
vF are propagation velocities for these free-fermions. The velocities were la-
beled differently to highlight the physical distinction for the symmetric combi-
nation of Ns =∑
rNr,s representing charge excitations and the antisymmetric
combination Js =∑
r rNr,s represents currents. This form of the Hamiltonian
is especially insightful because of the introduction of different velocity-like
variables vJ , vN , and how it transforms upon addition of interaction in the
model.
12 Motivation and Historical Context
The introduction of interactions does not change much in the derivation,
except that the bosons of choice in the Bogoliubov transformation are explicitly
interpreted as combinations of charge (ρ) and spin (σ) collective fluctuations.
The velocities of equation 2.7 are now altered by the coupling constants g2 and
g4 and the full TL Hamiltonian becomes
HTL =π~L
∑rνq 6=0
vν(q) : νr(q)νr(−q) : (2.7)
+π~2L
[vNν (N+ν +N−ν)
2 + vJν (N+ν −N−ν)2],
where ν = σ, ρ and the collective excitation velocity becomes
vν(q) =
√[vF +
g4ν(q)
π
]2
−[g2ν(q)
π
]2
, (2.8)
with vNνvJν = v2ν and vNν = vν/Kν . The conclusion of this derivation is that
interactions between the particle-hole pairs renormalize the velocity of the
collective excitations and the velocity (vρ) for charge propagation is no longer
the same as (vσ), the spin propagation velocity. This separation of two distinct
propagating modes is the well known “spin-charge separation”. The parameter
Kν is directly affected by the coupling constants and is sometimes described as
the stiffness of the 1D system because it regulates the decay of most correlation
functions. Because of its central role in describing the dynamics of excitations
within the TL model, it is often called the “Luttinger parameter”. Since we
will not work with a spin degree of freedom in the rest of this work, we will
drop the subscript and only write the Luttinger parameter as K.
2.2 Tomonaga-Luttinger Liquids 13
2.2.2 A Class of Luttinger Liquids
In his seminal contribution on this subject [9], Haldane recognized that the
construction of a harmonic-fluid description to solve electronic 1D systems
could be extended to a whole class of interacting particles in highly confined
configurations. In fact, the bosonization of 1D systems treats both fermions
and bosons similarly and the dynamics of the hydrodynamics of the low-energy
excitations can be described by the Hamiltonian derived above [11]. The long-
range density fluctuations emerge in all cases and their propagation is described
by the velocity v and the Luttinger parameter K. Of particular interest to us
is the case of spinless bosons interacting with realistic inter-atom potentials
[12]. Using the same derivation as the one shown in the previous section, we
can arrive at a very informative effective Hamiltonian [2]
H =~2π
∫ L
0
dz[vJ(∂zφ)2 + vN(∂zθ)
2], (2.9)
or equivalently
H =~v2π
∫ L
0
dz
[1
K(∂zφ)2 +K(∂zθ)
2
], (2.10)
where the velocity v =√vJvN defines the linearly dispersed density modes with
E(k) = ~vk and the two phases φ(z) and θ(z) are defined by the particle field
operator ψ† =√ρ(z)e−iφ(z). The angular field θ(z) is a suitable redefinition
of the density, such that
ρ(z) ≡[ρ0 +
1
π∂zθ(z)
] ∞∑m=−∞
ei2mθ(z).
14 Motivation and Historical Context
ρ0 is defined as the T=0 number density. ∂zθ(z) is the canonically conjugate
momentum to φ(z) and they must commute as [φ(z), ∂z′θ(z′)] = iπδ(z − z′).
The position z is set along the axis of the 1D system and the particle confine-
ment would be applied perpendicular to it. We will return to these equations
in section 2.5, where the convenient quadratic structure of the TL effective
Hamiltonian will allow for computation of the density-density correlations,
leading to predictions of physically measurable quantities.
In most 1D systems, the density correlation functions show a tendency to
form density waves, and in particular those correlation functions decay accord-
ing to power laws that are proportional to the interaction strength between
particles. These strong predictions of the similarity of behavior between sys-
tems of any quantum statistics and the power-law decay of physical quantities
as a function of interaction strength are key signatures for experiments to
observe [11]. A few strategies have been explored over the last two decades
to implement and measure such 1D systems. The most prominent ones are
described in the next section.
2.3 Realizations of 1D Systems
The interest in Tomonaga-Luttinger liquids (TLL) has been largely theoreti-
cal for decades and it is only recently that several techniques have allowed for
experimental realizations of quasi-1D systems in the laboratory. At its sim-
plest, a 1D quantum system can be created experimentally if test particles can
be confined along two axis and remain free to propagate in the third dimen-
sion. The application of this confinement potential gives rise to subbands in
2.3 Realizations of 1D Systems 15
the off-axis directions and if the energy of the particles is low enough, motion
perpendicular to the free axis is quenched. This remains true for different par-
ticles and for various confinement fields, as long as the energy of the particles
is lower than those transverse energy levels. There are therefore many physi-
cal systems where a realization of a 1D quantum systems could be attempted
and groups from several fields have indeed pursued it. For example, early
evidence of 1D behavior has been observed in electron gases within semicon-
ducting heterostructures [13, 14, 15, 16, 17], in optical lattices of cold vapor
gases [18, 19, 20, 21] and in the T-linear signature of the phonon heat capacity
of helium within porous materials [22]. An overview of some of the work done
in each of these avenues follows.
2.3.1 Electronic Transport in Quantum Wires
Electronic transport in condensed matter systems is a prime test bed for exotic
physical phenomena, and it is therefore no surprise that attempts to detect
signatures of TLL happened very early within this field. In particular, mea-
surements [13, 23] of the current tunneling into the edge state of the fraction
quantum Hall effect was found to be consistent with the presence of a chi-
ral Luttinger liquid. Realizations of TLL in electronic transport experiments
were also devised in systems where the conduction path was engineered to be of
very small dimensions. Technically, these are quantum wires where transport
is allowed in a limited number of conductance channels.
Using very narrow materials such as carbon nanotubes is one method of
obtaining quantum wires [24, 25] where 1D behavior could be observed. A
16 Motivation and Historical Context
second noteworthy approach uses confinement of electrons deep within semi-
conducting heterostructures. This strategy has evolved from the extensive
expertise of groups working on two-dimensional electron gases (2DEG), where
they can engineer stacks of carefully chosen semiconductors such that electrons
are free to move along two directions. Metal gates are then positioned over this
heterostructure such that applied voltages deplete the electronic density un-
derneath them, and conduction paths can thus be patterned precisely. Figure
2.3 shows an exampleof such a construction designed in our group .
Fig. 2.3 (A) Schematic of the 2DEG within a semiconductingheterstructure (not shown here) and the metallic gates used to de-pleted the 2DEG in chosen areas. This geometry combines twovertically integrated quantum wires to study the drag signal be-tween the two wires. (B) Scanning electron microscope image of adevice. Figure reproduced in part from [26].
Quantum wires made from 2DEGs have evolved into several parallel strate-
gies for the detection of signatures of the TLL. Resonant tunneling into a
quantum wire made from the cleaved-edge of a 2DEG [15, 17, 27] has been a
method of choice early on to observe TLL power-law behaviors. More recent
attempts are using a slightly different approach. By engineering two closely
spaced 2DEGs and placing gates such that a quantum wire is created in each
2.3 Realizations of 1D Systems 17
2DEG, the inter-wire interactions can be probed by measuring the Coulomb
drag signal between them [26, 28].
Figure 2.3A shows a schematic based on a device where two 2DEGs sep-
arated by tens of nanometers each have a pair of metallic gates to shape a
quantum wire. These two quantum wires are aligned vertically. Figure 2.3B
is an electron microscope image of an actual device where the metallic gates
(brighter areas) are clearly visible. These vertically integrated quantum wires
(shown in figure 2.3) have been used to demonstrate that the temperature
dependence of the Coulomb drag is consistent with having two parallel TLLs
[29]. Using similar experimental techniques, other groups have reported the
observation of charge fractionalization [30] and spin-charge separation [1, 31]
in quantum wires. A more detailed account of the experimental realization of
TLL with gated 2DEG systems can be found in the thesis of D. Laroche [32].
2.3.2 Laser Traps and Ultra Cold Atoms
Laser cooling and magneto-optical traps are experimental techniques that al-
low researchers to slow down atoms and cool them to extremely low temper-
atures. The optical trapping technique uses a retro-reflected laser beam that
interferes with itself to create a standing wave. The atoms develop an induced
polarization when interacting with this electric field, and consequently feel a
force proportional to its gradient. The laser beam frequency is also tuned with
respect to a specific atomic transition such that many photons interact with
the atoms and the latter can eventually aggregate in a spacial lattice formed
by the standing light pattern. If three such counter-propagating laser beams
18 Motivation and Historical Context
are placed perpendicular to one another, then a 3D lattice is created where a
dilute gas of atoms can be trapped.
The optical trap can be used in conjunction with a magnetic trap, where
the spin of the atoms interacts with an applied magnetic field through the
Zeeman effect. Various methods [20, 33] can be used here to magnetically
confine the atoms, but the goal remains to trap the cooler atoms while letting
the warm ones escape to the vacuum surrounding the trap.
VOLUME 87, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 15 OCTOBER 2001
Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates
Markus Greiner, Immanuel Bloch, Olaf Mandel, Theodor W. Hänsch,* and Tilman EsslingerSektion Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4/III, D-80799 Munich, Germany
and Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany(Received 4 May 2001; published 1 October 2001)
Bose-Einstein condensates of rubidium atoms are stored in a two-dimensional periodic dipole forcepotential, formed by a pair of standing wave laser fields. The resulting potential consists of a latticeof tightly confining tubes, each filled with a 1D quantum gas. Tunnel coupling between neighboringtubes is controlled by the intensity of the laser fields. By observing the interference pattern of atomsreleased from more than 3000 individual lattice tubes, the phase coherence of the coupled quantum gasesis studied. The lifetime of the condensate in the lattice and the dependence of the interference patternon the lattice configuration are investigated.
DOI: 10.1103/PhysRevLett.87.160405 PACS numbers: 03.75.Fi, 03.65.Nk, 05.30.Jp, 32.80.Pj
The physics of Bose-Einstein condensation is governedby a hierarchy of energy scales. The lowest energy is usu-ally the atomic oscillation frequency in the trap which ismuch smaller than the chemical potential of the conden-sate. Here we report on experiments in which we enter aregime where this order is inverted. By overlapping twooptical standing waves with the magnetically trapped con-densate we create a two-dimensional periodic lattice oftightly confining potential tubes. In each of the severalthousand tubes the chemical potential is far below the trap-ping frequencies in the radial direction. The radial motionof the atoms is therefore confined to zero point oscillations,and transverse excitations are completely frozen out. Inthe degenerate limit, these 1D quantum gases are expectedto show a remarkable physics not encountered in 2D and3D, for instance, a continuous crossover from bosonic tofermionic behavior as the density is lowered [1–5].
In our two-dimensional periodic array of quantum gasesthe tunnel coupling between neighboring lattice sites iscontrolled with a high degree of precision by changingthe intensity of the optical lattice beams. A similar con-trol over coupling between adjacent pancake-shaped con-densates was achieved in recent experiment using a singlestanding wave laser field [6,7]. After suddenly releasingthe atoms from the trapping potential we observe the mul-tiple matter wave interference pattern of several thousandexpanding quantum gases. This allows us to study thephase coherence between neighboring lattice sites, whichis remarkably long lived. Even for long storage times,when the phase coherence between neighboring latticesites is lost and no interference pattern can be observedanymore, the radial motion of the atoms remains confinedto zero point oscillations.
Similar to our previous work [8], almost pure Bose-Einstein condensates with up to 5 3 105 87Rb atoms arecreated in the jF ! 2, mF ! 2! state. The cigar-shapedcondensates are confined in the harmonic trapping poten-tial of a QUIC trap (a type of magnetic trap that incor-porates the quadrupole and Ioffe configuration) [9] withan axial trapping frequency of 24 Hz and radial trapping
frequencies of 220 Hz. The lattice potential is formed byoverlapping two perpendicular optical standing waves withthe Bose-Einstein condensate as shown in Fig. 1. All lat-tice beams are derived from the output of a laser diodeoperating at a wavelength of l ! 852 nm and have spotsizes w0 (1"e2 radius for the intensity) of approximately75 mm at the position of the condensate. The resultingpotential for the atoms is directly proportional to the in-tensity of the interfering laser beams [10], and for the caseof linearly polarized light fields it can be expressed by
U# y, z$ ! U0% cos2#ky$ 1 cos2#kz$1 2e1 ? e2 cosf cos#ky$ cos#kz$& . (1)
Here U0 describes the potential maximum of a singlestanding wave, k ! 2p"l is the magnitude of the wavevector of the lattice beams, and e1,2 are the polarizationvectors of the horizontal and vertical standing wave laserfields, respectively. The potential depth U0 is convenientlymeasured in units of the recoil energy Er ! h2k2"2m,with m being the mass of a single atom. The time-phasedifference between the two standing waves is given by the
FIG. 1. Schematic setup of the experiment. A 2D lattice poten-tial is formed by overlapping two optical standing waves alongthe horizontal axis ( y axis) and the vertical axis (z axis) with aBose-Einstein condensate in a magnetic trap. The condensate isthen confined to an array of several thousand narrow potentialtubes (see inset).
160405-1 0031-9007"01"87(16)"160405(4)$15.00 © 2001 The American Physical Society 160405-1
Fig. 2.4 Schematic representation of a 2D optical lattice. Thecloud of cold gas is formed with perpendicular lasers beams, hereshown as arrows, that create the electric fields forcing the trappingof atoms. (BEC stands for Bose-Einstein condensate). The insetshows how the cloud of atoms is divided in highly asymmetric, elon-gated “cigar” shaped regions arranged in a 2D lattice. Each “cigar”contains a few thousand atoms that are in a quasi-1D confined state.Reproduced from [18].
Figure 2.4 shows a simplified illustration of the vapor cloud created using
magneto-optical traps. Within a vacuum chamber, the cold atoms are confined
to the volume at the intersection of the laser beams (here shown as arrows). In
this case, the intensity and frequency of the lasers was chosen to create a highly
asymmetric trap such that atoms would be confined in a 2D lattice of smaller,
2.3 Realizations of 1D Systems 19
elongated “cigar shaped” traps. The few thousand atoms in each “cigar” are
highly confined radially and are assumed to be in a quasi-1D lattice system.
The possibility of using bosonic atoms and the inherent low temperatures
of such cold gas clouds offers the opportunity to investigate 1D systems of
interacting bosons [18, 19, 21] and explicitly test if the TLL description applies.
One key feature of magneto-optical traps is the ability to vary the potential
depth that the atoms experience by modulating the intensity and frequency
of the laser beams forming the lattice of potentials. It is thus possible, in
principle, to effectively tune the interaction strength between atoms sitting in
adjacent potential wells and, presumably, be able to observe the dynamics of
1D systems [34] for different Luttinger parameters and compare the results to
TLL theory [33, 35, 36].
The tunability of interaction between the constituent particles of cold va-
por gases is particularly effective when making a comparison between the
many variations of microscopic models [12], such as the hard-core bosons
of the Tonks-Girardeau gas or the generic Bose-Hubbard particles in a lat-
tice. As opposed to electronic systems, the nature of inter-atom interaction
is not Coulombic and is better described as a short-range interaction leading
to weakly coupled system. The very nature of dilute gases imposes a limit
on the density of particles accessible and the total sample size is restricted to
mesoscopic dimensions. Realizations of bosonic 1D systems with higher den-
sities, strong interactions and Galilean invariant (i.e. without the underlying
lattice structure of cold gases) are still lacking [2]. A candidate substrate for
the latter requirements is liquid helium confined in nanopores.
20 Motivation and Historical Context
2.3.3 Helium Adsorbtion in Porous Media
In the quest for experimental realizations of TLL systems, some groups have
investigated the adsorption of atoms inside and outside axi-symmetric porous
materials. Porous materials such as zeolites can be chemically tailored to form
large arrays of stacked tubes with diameters a only a few nanometers. These
nanotubes can be very long and have fixed diameters since their chemical
composition is the key factor determining their dimension. It is a favorable
characteristic of these materials to be able to confine particles over long dis-
tances and thus, presumably, preserve the long-range properties of 1D systems.
LowTe
Inner-tube sites
interstitial sites
surface sites
Fig. 2.5 Cartoon of zeolites used in adsporption experiments.The curled-up sheets form bundles of nanotubes with approximatehexagonal stacking. In this drawing, the length-to-diameter ra-tio is not representative of actual ratios in typical samples, andis depicted here in a very regular and ordered stack, which is anidealization of the actual structure of dry zeolite powder.
Zeolites can be readily synthesized in bulk and transformed to a dry pre-
cipitate of fully formed nanotubes (see cartoon in figure 2.5). This powder can
then be used in experiments to adsorb atoms on the interstitial sites between
adjacent nanotubes and, as is hoped, also within the core of the nanotubes
2.3 Realizations of 1D Systems 21
once all other adsorbtion sites are filled. Torsional oscillator studies of helium
atoms adsorbed on such nanotube bundles have been performed and promising
results of quasi-1D behavior have been obtained measuring the phonon dissipa-
tion in various zeolite geometries [37, 38]. Once again, it is worth mentioning
that since these experiments are done with helium atoms as a test particle and
4He is a boson, these experiments are complementary to those performed in
electronic quantum wires.
It is important to note that experiments with zeolites are performed with a
considerable volume of crushed nanotubes powder. This means that a sample
consists of a very large total number of nanotubes, since each bundle can
contain any number of them, and those nanotubes are of various dimensions,
since each bundle differs greatly in length or width. Another challenge of using
channels with a large distribution of sizes is that it is not immediately clear
what is the ratio between adsorption sites on the surface of tubes and those
sites of interest, where 1D confinement might be observed. Compounding this
difficulty is the fact that sites on the surface of nanotubes are populated first
and vastly outnumber the “potentially-1D” sites such as the groove between
two nanotubes (interstitial sites) or inside the nanotube core (inner-tube sites)
[39]. Any signal measured from a powder with zillions of tubes will always have
a strong component emanating from the two-dimensional surfaces of both the
experimental cell and the nanotube’s outer surface. It is therefore challenging
to extract results specific to the atoms in a quasi-1D state.
The intuitive solution to these challenges is to use a single nanotube. While
the ratio of relevant signal over the overall measured signal would be much
22 Motivation and Historical Context
more favorable in this case, the measurement techniques used in the torsional
oscillator experiments would need to be several orders of magnitude more
sensitive and it is therefore necessary to adopt another strategy.
2.4 Helium as a Paradigm
It is in the context of the emergence of the experiments presented in the last
section that we attempted to devise a new experimental scheme that would al-
low us to avoid measurements averaging over a large distribution of nanotubes,
but still have the sensitivity to detect signal from helium atoms propagating
inside a single nanotube. The use of strongly-interacting particles, such as
helium atoms in the liquid phase, is also an asset to complement the laser-trap
experiments that use dilute gases. Finally, the ability to repeat an experiment
with either fermions or bosons is a powerful tool to verify the predictions that
Tomonaga-Luttinger liquids behavior is independent of quantum statistics.
The two isotopes 4He and 3He are thus ideally suited for this purpose.
Indeed, helium has been a model system of choice for experiments carried
out over a very broad range of conditions and its thermodynamic properties
are available at all temperatures and pressures of interest here. Figure 2.6
shows the pressure-temperature phase diagram of 4He with semi-logarithmic
axes. The region labeled “He I” represents the liquid phase.
The large zero-point motion of the helium atoms prevents it from ever so-
lidifying at low pressure (< 25 atm), which gives it the unique potential to
transition to a quantum mechanically dominated phase at low temperatures:
the superfluid phase. This second order phase transition to superfluidity is
2.4 Helium as a Paradigm 23
0102
103
Pre
ssur
e [P
a]
Temperature [K]
104
105
106
107
Pre
ssur
e [b
ar]
10-3
10-2
10-1
1
101
102
1 2 3 5 6 74
Gas
He I
He II
Solid
criticalpoint
Fig. 2.6 Pressure-temperature phase diagram of bulk 4He. Theliquid phase is labeled “He I” and is bounded near 2.17 K by a nearlyvertical phase transition line called the “λ-line”. The transition overthe λ-line leads to the “He II” region where superfluidity emerges.A unique property of helium is the absence of the solid phase atlow temperature for pressures below ∼25 atm.
bounded in the P-T diagram by a line quizzically dubbed the “λ-line” due to
the similarity between the shape of the 4He specific heat curve and the Greek
letter λ. The λ-line, and λ-temperature, are especially relevant in investiga-
tions of superfluids. For example, experiments with porous materials such
as Vycor and Gelsil [40, 41, 42] found a suppression of superfluidity causing
a lowering of the λ-line as a function of pore size. Given these characteris-
tics, helium atoms are paradigm particles to study fluid transport at very low
temperatures in highly confined systems [43, 44].
24 Motivation and Historical Context
2.5 Quantum Monte Carlo Simulations
Our search for a suitable experimental system optimized to study strongly in-
teracting one-dimensional states of matter is further supported by large-scale
numerical simulations of 4He filled nanopores [45]. In this work and in an
earlier publication [46], Del Maestro and Affleck show results on large-scale
quantum Monte Carlo (QMC) simulations of 4He atoms exhibiting behav-
ior consistent with TLL theory. The simulations used the path integral Monte
Carlo method and a “Worm Algorithm” to efficiently explore the paths (world-
lines) of each quantum particle evolution during the simulation. The model is
built from a generic many-body Hamiltonian within an external potential Vext
and inter-particle interactions Vint,
H =N∑i=1
[ −~2
2MHe
~∇2i + Vext(~ri)
]+∑i<j
Vint(~ri − ~rj), (2.11)
where the position of each of the N atoms is ~ri and they have mass MHe. The
symbol h = 2π~ represents Planck’s constant. In the latest series of simula-
tions, great care was given to the choice of Vint and Vext, and in particular,
an Si3N4 boundary was selected to match as closely as possible the experi-
mental conditions of the work in this thesis. The results of these simulations
are considered “exact” in the sense that realistic particle-particle (Vint) and
particle-wall (Vext) interaction potentials are used and they can be computed
for experimentally accessible pressures and temperatures.
One interesting finding from the averaged particle distribution within the
nanopores is the formation of concentric shells as shown in figure 2.7. These
2.5 Quantum Monte Carlo Simulations 25
cylindrical shells start forming near the nanopore wall, where a large potential
induces the formation of a layer of atoms at a distance determined by the min-
imum in the wall-particle interaction energy. The next layer forms when atoms
settle on the first layer at a distance given by the helium-helium interaction
potential. This effectively forms an inner cylindrical shell. If the radius of the
nanopore is large, many layers can fit concentrically. For multiples of ∼3 A,
simulations show that there is sufficient space for an inner core to form in the
middle of the nanopore, as can be seen in the insets of each graph in figure
2.7. For example, with a radius of 6 A, there is one cylindrical sheet of atoms
surrounding an inner core. The atoms located within that inner core were, in
fact, found to be the best candidates to study the 1D system.
2.5.1 Comparison to the Luttinger Liquids Model
The results of the QMC simulations are then compared with predictions of the
Tomonaga-Luttinger Liquid (TLL) model. Writing the bosonic many-body
Hamiltonian of equation 2.11 in second quantized notation can help one see
the emergence of the 1D signatures of the TLL model. We get
H =
∫ L
0
dz
[~2
2MHe
∂zΨ†(z)∂zΨ(z) +
1
2
∫ L
0
dz′ρ(z)V1D(z − z′)ρ(z)
], (2.12)
where the bosonic creation operator Ψ†(z) =√ρ(z)e−iφ(z) has the usual com-
mutation relation [Ψ(z),Ψ†(z′)] = δ(z − z′). We recognize the presence of
the density operator ρ(z) and the phase operator φ(z), and using the tools of
section 2.2, we can easily return to the Hamiltonian shown in equation 2.9:
26 Motivation and Historical Context
July 20, 2012 9:1 WSPC/Guidelines-IJMPB S021797921244002X
A Luttinger Liquid Core Inside Helium-4 Filled Nanopores
4 6 8 10 12Pore Radius [A]
0.010
0.012
0.014
0.016
0.018
0.020
0.022
Volu
me
Den
sity
[A−
3 ]
T = 0.50 K
Fig. 3. The average volume density as a function of nanopore radius for helium at SVP andT = 0.5 K.
0!00
0!05
0!10
0!15
0!20
0!25
Rad
ial
Den
sity
A"
3 R = 2.9 A R = 4.0 A
0!00
0!05
0!10
0!15
0!20
0!25
Rad
ial
Den
sity
A"
3 R = 6.0 A R = 8.0 A
0 2 4 6 8 10r A
0!00
0!05
0!10
0!15
0!20
0!25
Rad
ial
Den
sity
A"
3 R = 10.0 A
0 2 4 6 8 10r A
R = 12.0 A
Fig. 4. The radial number density ρR(r) of helium inside nanopores with radii R = 2.9, 4.0, 6.0,
8.0, 10.0, 12.0 A with an inset showing a full instantaneous worldline configuration inside the pore
measured in the QMC simulations projected on z = 0. All pores of L = 100 A and are held at
SVP corresponding to µ = −7.2 K.
1244002-7
Int.
J. M
od. P
hys.
B 2
012.
26. D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific.
com
by M
CG
ILL
UN
IVER
SITY
on
03/1
9/14
. For
per
sona
l use
onl
y.
Fig. 2.7 Radial density of helium atoms inside nanopores of ra-dius between 2.9 and 12 A. Insets shows atoms on a black back-ground seen looking along the nanopore axis. Reproduced from[2].
HTLL =~2π
∫ L
0
dz[vJ(∂zφ)2 + vN(∂zθ)
2]. (2.13)
The transformation of the interacting 1D boson Hamiltonian into a quadratic
form has the immediate benefit that many thermodynamic quantities can be
computed [2] from a very limited number of input parameters such as the
temperature T , the propagation velocity v and the Luttinger parameter K.
2.5 Quantum Monte Carlo Simulations 27
For example, density correlations can be computed for the atoms within the
inner cores and compared to predictions from TLL theory as seen in figure
2.8. The results of the fit of the Luttinger liquid theory to the QMC data
show a strong tendency to form ordered density waves along the length of the
nanopore for the smallest pores. The fit yields a value for the Luttinger pa-
rameter K = 6.0(2), a value associated with a system with low interactions
and a tendency to localize and develop density wave order (figure 2.8a).
Fig. 2.8 Density correlation of helium atoms along the channelaxis for a nanopore of radius R=2.9 A (a) and R=12 A (b). Thedecay of the long range order is characterized by an interaction thatcorresponds to Luttinger parameters K= 6.0 and 1.3 respectively.Curves are manually offset for clarity with the bottom data simu-lated at T=0.5 K increasing up to 2.0 K for the data at the top.Reproduced from [2].
The same analysis applied to the larger pore with radius 12 A shows
damped oscillations possibly due to inter-layer coupling increasing the effec-
tive interactions between atoms of the core. The increased effective interac-
28 Motivation and Historical Context
tions between particles in the core channel are characterized by a fit parameter
K = 1.3(2) which describes a system incrementally dominated by superfluid
fluctuations (see figure 2.8B). The smooth crossover from a phase dominated
by fluctuations to one where density wave ordering is present is expected in
TLL theory. The possibility of tuning the interaction strength by testing differ-
ent pore sizes is thus an important asset to probe this crossover and determine
whether or not a quantum phase transition occurs.
One important conclusion drawn from the results of density wave order-
ing as the pores decrease in size is that there is a decrease of the effective
superfluid fraction within the pore correlated to the rise of density wave or-
der, which may be observed experimentally. Indeed, experiments of transport
through nanotubes are very sensitive to the emergence of superfluidity in liq-
uid helium and represent, historically, a prominent method to determine the
superfluid fraction of helium under different conditions (see for example review
of superfluid helium in porous media [40]). As an example, Del Maestro has
calculated in [2] that at 1.0 K the superfluid fraction inside the nanopore is
close to 20%, where the vast majority of superfluid is in the inner core.
The measurement of the superfluid fraction within nanopores is particu-
larly relevant to the experiments we developed in this thesis. Recent work
by Kulchytskyy et. al [3] shows that QMC computation of helium within
nanopores has a unique superfluid signature. Figure 2.9 shows the temper-
ature dependence of the superfluid fraction inside nanopores. The absolute
fractions were renormalized by introducing an ad hoc cut-off radius to ex-
clusively count the atoms within the inner core. The inset shows the phase
2.5 Quantum Monte Carlo Simulations 29
velocity vJ found in the TLL model of equation 2.13 as a function of the radius
of nanopores.LOCAL SUPERFLUIDITY AT THE NANOSCALE PHYSICAL REVIEW B 88, 064512 (2013)
0.00
0.01
0.02
0.03
0.04
ρW s
(r)
[ A−
3]
T = 0.75 KT = 1.00 KT = 1.25 K
0 2 4 6 8 10 12
r [A]
0.00
0.01
0.02
0.03
0.04
ρA s(r
)[ A
−3]
R = 13.0 A
FIG. 4. (Color online) The temperature dependence of the localsuperfluid density measured via the winding number (top) and area(bottom) estimator for a nanopore with radius R = 13.0 A and lengthL = 75 A. The shaded region in the background corresponds to theparticle density ρ at T = 0.75 K.
solid (K ≫ 1) order. For a real physical system, the velocitiesvJ ≡ v/K and vN ≡ vK can be related to the parameters of theunderling many-body Hamiltonian. By comparing the predic-tions of harmonic LL theory, derived from the grand partitionfunction Z = Tr exp[−β(H − µN )] with the measurementsfrom finite temperature QMC simulations, vJ and vN can bedetermined. For quasi-1D helium confined inside nanoporeswith R < 3 A, this has already been accomplished,26 but forlarger radius pores, required the use of an ad hoc cutoff radiuswhen analyzing QMC data. The physical origin of this cutoffis now fully understood as the radius of the superfluid core,and we expect it to be described by LL theory:27
ρWs
ρc
= 1 − πhβvJ
L
!!!!θ ′′
3 (0,e−2πhβvJ /L)θ3(0,e−2πhβvJ /L)
!!!! , (14)
where θ3(z,q) is a Jacobi theta function withθ ′′
3 (z,q) ≡ ∂2z θ3(z,q) and ρc = (Nc/N )ρ where Nc is
the number of atoms in the core. For each radius, wehave performed a rescaling of the total superfluid responsedisplayed in Fig. 2 and determined the velocity vJ (R) througha fitting procedure that yields the best collapse of all lowtemperature data onto Eq. (14). The results are displayedin Fig. 5 where the temperature scaling of the nanoporesuperfluidity is consistent with Luttinger liquid theory.
Much remains to be done, including confirming thepredicted pore length scaling of ρW
s /ρc and evaluating theR-dependent LL parameter K . In addition, it seems naturalto contemplate the effects of disorder, surely present in thepore walls, as well as the introduction of fermionic 3He which
0 1 2 3 4 5 6
L/(hβvJ )
0.0
0.2
0.4
0.6
0.8
1.0
ρW s
/ρc
R = 10
R = 11
R = 12
R = 13
R = 14
R = 15
10 11 12 13 14 15
R A
0
50
100
150
hv J
AK
FIG. 5. (Color online) The superfluid fraction of the core ofnanopores for varying radii which can be collapsed onto the universalprediction from Luttinger liquid theory. The inset shows the extractedvalue of the phase velocity hvJ obtained by fitting to the windingnumber estimator for each radius.
may strongly alter superfluidity as bosonic exchanges will besuppressed in one dimension.
V. CONCLUSION
We have performed large-scale quantum Monte Carlo simu-lations for helium-4 confined inside short 75 A pores with radiibetween 1–1.5 nm. The results show a finite and anisotropicsuperfluid response above T = 0.5 K, with a magnitude thatis dependent on whether longitudinal or rotational motionof the nanopore is considered. The difference is large andarises from the absence of any classical moment of inertia inthe truly 1D limit where flow is still possible. Experimentsprobing this remarkable breakdown of the two-fluid picturecould be performed by comparing the superfluid fractionmeasured by capillary flow and a nanoscale Andronikashvilitorsional oscillator. Our results also indicate that when theradii of nanopores becomes sufficiently small, the superfluidfraction may exhibit plateaus, increasing in steps, due tothe classical sticking of wetting layers near the pore walls.This is in stark contrast to the usual smooth temperaturedependence of ρs/ρ observed for bulk 4He and could providea signature of the crossover to 1D behavior. If the fraction ofatoms adhering to the nanopores’ walls could be discerned,possibly by comparing flow rates at high and low temperature,an examination of the finite size and temperature scalingof the superfluid density would confirm that confined low-dimensional helium is a Luttinger liquid. This would open upan exciting strongly interacting and high density regime wherethe effective low energy theory can be experimentally testedin systems with Galilean invariance.
ACKNOWLEDGMENTS
We acknowledge financial support from the Universityof Vermont, NSERC (Canada), FQNRT (Quebec), and theCanadian Institute for Advanced Research (CIFAR). Thisresearch has been enabled by the use of computationalresources provided by the Vermont Advanced ComputingCore supported by NASA (NNX-08AO96G), WestGrid,SHARCNET, and Compute/Calcul Canada.
064512-5
Fig. 2.9 Superfluid fraction inside the core (subscript “c”) ofnanopores of various radii R as a function of temperature (β ≡1/kBT ). The nanopores in the simulation had length L and ra-dius R. Inset shows the phase velocity extracted from a fit to TLLtheory. Reproduced from [3].
Again, the good agreement of the QMC simulations with the predictions of
TLL theory (solid grey line) for atoms confined within the nanopore speaks to
the potential of observing 1D signatures in liquid helium transport measure-
ments. In other words, for a sufficiently narrow and long transport channel
at sufficiently low temperature, the simulation results predict that 4He atoms
should behave as a strictly 1D system with the nature of the inter-particle
interactions well described by the TLL theory.
Chapter 4 will describe how we integrated some of the strengths and al-
leviated some of the drawbacks of existing experiments to develop a new ex-
perimental scheme in the search for an experimental realization of Luttinger
liquids. Prior to the exploration of the experiment design, a chapter is devoted
30 Motivation and Historical Context
to present an overview of the theoretical foundations required to interpret the
data in the thesis.
31
Chapter 3
Theory of Mass Flow in Short
Pipes
The physics of fluidic systems has attracted the interest of generations of scien-
tists and engineers both for the breadth of physical phenomena it encompasses
and for the critical knowledge that modelization provides to several industries.
In fact, key insights into the physics of fluids can be obtained by making a
few assumptions, however, the Navier-Stokes equations are complex and still
famously carry a Clay Mathematics Institute prize for the generic proof of their
solvability. The experiments I present in the next chapters are conducted over
a wide range of thermodynamic conditions and span different regimes of mass
transport. Since the dynamics of fluids are strongly affected by the physical
characteristics of the fluid, several simplified models are required to explain
the fluid behavior over the whole thermodynamic range. For example, the
assumptions built in a model for low-density fluids are not necessarily valid at
32 Theory of Mass Flow in Short Pipes
higher density and a new model must be used instead.
3.1 Knudsen Number As an Indicator of Flow Regimes
A good indicator of the appropriate model to choose to describe the flow in
a given system is the Knudsen number Kn. This dimensionless parameter is
defined as the ratio of the mean-free-path (λ) of a particle to the characteristic
length scale of the environment in which it propagates. In the case of mass
transport through circular apertures, the convention is to compare the mean-
free-path to the diameter (D) of the pore.
Fig. 3.1 The Knudsen number scale indicates the regions of ap-plicability of different fluid dynamics models.
3.1.1 Mean-Free-Path of a Gas
The mean-free-path is the distance a particle can travel between interactions,
averaged over the entire velocity distribution of the particles in the system.
Given a gas with a Maxwellian distribution of velocities and concentration n
in units of atoms per cubic meter, the mean-free-path is:
3.2 A Continuum of Flow Regimes 33
λ =1√
2nπσ2. (3.1)
Here, the factor of√
2 comes from averaging collisions between particles
with a Maxwell distribution of velocities and σ is the collision cross section.
This latter equation assumes an homogeneous gas at equilibrium and, using
the ideal gas law1, n = PkBT
, we obtain
λ =kBT√2πσ2P
. (3.2)
The symbols kB, T and P are the Boltzmann constant, the thermodynamic
temperature and pressure respectively. Finally, the cross section diameter is
well known for helium and is equal to 2.18 A (table 1.2-2 in [47]).
3.2 A Continuum of Flow Regimes
3.2.1 Knudsen Effusion of Simple Gases
A statistical approach can be used to treat gas particles interacting only
through elastic collisions if they have a well defined distribution of velocities.
Taking an arbitrary surface A and integrating, over the whole solid angle, all
velocity vectors of particles with a Maxwell-Boltzmann distribution [48], one
finds an average flux through a given surface:
1The ideal gas law is, in principle, only accurate at low gas densities. Given that we onlyuse the ideal gas law to find the mean-free-path λ, which gives us the Knudsen number Kn,any deviation of the physical mean-free-path at high-density from equation 3.2 is not prob-lematic in our modeling. The small error in our calculated λ only modifies the position ofdata on the abscissa. The flow computed from theoretical models is also displayed for Kncomputed with equation 3.2 and the fit to the data is therefore not affected by this smalloffset.
34 Theory of Mass Flow in Short Pipes
Qpinhole =1
4nvA. (3.3)
Here, the solution is expressed with the average particle velocity v =√
8kBTπM
,
where M is the mass of the particles. If one considers an enclosure with a
single opening and defines that opening as the surface A, then the flux of
atoms out of the enclosure is given by equation 3.3. We obtain the mass flow
through an aperture of area A by substituting the average velocity v and the
atom concentration n in equation 3.3 such that
Qpinholem = Qpinhole ·M
=
(1
4nvA
)·M
=1
4
(P
kBT
)(√8kBT
πM
)(πR2) ·M
=
(R2
√πM
2kBT
)· P ≡ Gpinhole · P (3.4)
and R is the radius of a circular aperture. Here, the subscript “m” denotes
a flow in units of kg/m3 and Gpinhole is the conductance of a pinhole in the
Knudsen effusion regime when a pressure head P is applied.
Gas Flow in Short Pipes
The assumption of a pinhole geometry is an approximation valid only when
the pore diameter is much larger than the pore length (D L). Solid-state
nanopores have dimensions that are typically closer to D ≈ L and the walls
cannot be ignored in the calculation of the pore conductance. Intuitively, one
3.2 A Continuum of Flow Regimes 35
can imagine how collisions with a pore wall decrease conductance compared
to the pinhole case. These collisions are diffusive and can cause particles
to bounce back towards their origin. Longer pores increase the probability
of collisions and thus reduce the conductance of the pore. Figure 3.2 shows
a cartoon representation of this situation, where a longer pore causes more
collisions with the walls and leads to a reduced total conductance.
Fig. 3.2 Side view of transport channels with particles diffusingthrough. The conductance of the pinhole (A) is larger than theshort (B) and long(C) pipes because longer pores increase the oddsof back reflections due to collisions with the walls.
In fact, the mass transport through a channel of finite length at high Knud-
sen number can be modeled using a transmission probability T . Both the
length of the pore and the nature of the collisions with the wall affect T . As-
suming diffusive collisions at the walls allows one to compute the probability
that a particle passes from one side of a channel to the other, accounting for
all collisions along the way.
36 Theory of Mass Flow in Short Pipes
The Clausing Factor
The effective conductance of the pore is then reduced by the transmission
probability to
G ≡ T Gpinhole,
where Gpinhole is the conductance of the entrance of the channel. Clausing
first introduced this concept for the effusion of gases through channels and he
defined a factor K [49], latter named the “Clausing factor”, to account for the
reduced conductance. This Clausing factor takes into account the probability
of transmission from every cross sectional “slice” along the tube and adds
them in series. For cylindrical short tubes, approximate equations have been
found [50], and simulations performed [51] such that the total transmission
probability can be computed. The solution from Berman [50] is
K = 1 + y2 − y√y2 + 1−
[(2− y2)
√y2 + 1 + y3 − 2
]2
4.5y√y2 + 1− 4.5ln
(y +
√y2 + 1
) , (3.5)
where y ≡ L/(2R) for simplicity.
The Clausing factor varies from perfect transmission of 1.0 to a value of
8R3L
as the length-to-radius ratio goes from zero to infinity. In addition, it has
been shown [52] that the Clausing factor in pores with elliptical cross-sections
has a correction of less than 2% with respect to that of circular pores of the
same area. This indicates that small deviations of nanopore cross section
from a perfect circle should have little discernible effect on the mass transport
3.2 A Continuum of Flow Regimes 37
modeling.
0.2
Cla
usin
gfa
ctor
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Ratio L/R
50o
30o
20o
10o
5o
1o
0o
Eq. 3.5
0.1 1 10
θ
Fig. 3.3 Clausing factor computed by Iczkowski [53] (circles) fora series of angles of the opening of a transport channel. A perfectcylindrical tube has an angle of 0 and a pinhole has an angle of90. The dotted lines are guides-to-the-eye and the solid black lineis calculated from equation 3.5.
In the earliest computational studies of diffusive transport, the cross section
of the channels was kept constant along the length of the channel. This turns
out to be a very good approximation for large experimental systems. However,
it is not the case for our nanopores since there is usually a ∼10 opening angle
between the wall and its axis. Therefore, the constant cross section assumption
may no longer be appropriate to model the flow in the smaller pores that we
studied. Recent computational work has however been done [53, 54, 55] to
account for a non-constant cross section where diffusion in channels with a
38 Theory of Mass Flow in Short Pipes
conical shape were simulated and the dependence of the Clausing factor on the
opening angle was extracted. Figure 3.3 shows the Clausing factor for angles
ranging from 0 to 50 as a function of the length-to-radius ratio (L/R)[53].
The angle used in the computation is measured between the pore wall and the
pore axis, as seen in the diagram inside figure 3.3. Once the Clausing factor is
determined, we can then compute the expected conductance of a pore in the
Knudsen effusion regime, which is given by
GKnudsen = K
(R2
√πM
2kBT
). (3.6)
3.2.2 Hydrodynamics of Viscous Flow
The typical mean-free-path of helium at standard pressure and temperature
(STP) is, from equation 3.2, λ ≈ 10−6 m, which is comparable to the nanopore
dimensions. This makes it experimentally possible to access different flow
regimes by a simple tuning of the mean-free-path via a change in pressure.
It is usually assumed that mass flow in systems with Kn. 10−2 are in the
continuum regime, and that small corrections accounting for slip behavior at
the wall occurs for 10−2 . Kn . 10−1. The regime that bridges the gap from
Knudsen effusive flows to viscous flows is referred to as the transitional regime.
We can model viscous fluid dynamics by making three hypotheses regarding
the fluid. The first is mass conservation, where by virtue of the continuity
equation, a fluid element is preserved in space and time from one position to
the next. The second is momentum conservation of the fluid which regulates
how forces modify the velocity field. The last one is, as expected, energy
3.2 A Continuum of Flow Regimes 39
conservation. These three assumptions, once combined, give the Navier-Stokes
equation of fluid dynamics. This equation is, however, hard to solve in general
and further assumptions must be made to obtain an analytic equation for the
flow.
As stated in the section treating Knudsen effusion, the nanopore has a
length-to-radius ratio close to unity (LR≈ 1). This prevents us from using the
“infinite-length” approximation that gives rise to the well-known Poiseuille
flow in long channels,
Q∞viscous =πR4∆P
8ηL, (3.7)
where η is the dynamic viscosity and ∆P is the pressure measured across the
pipe. Langhaar [56] has used a linearization of the Navier-Stokes equation to
describe the pressure drop in the transition length of a straight tube. In a finite
pipe, an additional pressure drop is caused by the acceleration of the fluid at
the ends of the channel where a fully-developed flow profile is not yet reached,
i.e. in the transition length. This additional contribution is proportional to a
factor α, and so the pressure drop is given by
∆P =32ηLv
D2+α
2ρv2, (3.8)
where D is the channel diameter. Solving for the velocity v in the latter
equation gives
v =32ηL
αρD2
(√1 +
αρD4
512η2L2∆P − 1
), (3.9)
40 Theory of Mass Flow in Short Pipes
and the mass flow is given by Qm = ρvA, where ρ is the density and the area
A is taken as a circle of radius R ≡ D/2. The viscous flow in a pore where the
transition length is taken into account is thus given by
Qm =8πηL
α
(√1 +
αρR4
32η2L2∆P − 1
). (3.10)
This equation will be further discussed throughout chapter 5.
Reynolds number in Nanopores
Hydrodynamic equations are often described in terms of dimensionless num-
bers that help separate systems into distinct groups. The Knudsen number
used in this chapter is an example of a dimensionless number that classifies
different flow regimes. Another useful dimensionless parameter is the Reynolds
number, which compares the effect of viscosity to typical system dimensions
and thus informs us on diffusive behavior, such as the onset of turbulence. The
Reynolds number is defined as
Re =ρvD
η,
for cylindrical pores. Once converted using the definition of mass flowQm = ρvA,
it readily gives
Re =4Q
πDη. (3.11)
Given the non-linear dependence of flow on the diameter of the pore, the
highest Reynolds number for gas flow will be obtained in pores with large
3.3 Liquid Helium Flow 41
diameters and at low temperatures where the viscosity is smallest. In partic-
ular, for the ∼100 nm sample presented in chapter 5, the Reynolds number is
estimated to be at most Re ≈ 4·3x10−11kg/sπ·100x10−9nm·8.8x10−6Pa·s = 43 at liquid nitrogen
temperature and for the highest pressures reached. At the opposite end, small
nanopores with diameters of approximately 15 nm have Re ≈ 0.14 at room
temperatures. The same calculation can be accomplished for the liquid helium
flow experiments with the D = 45 nm pore, where the mass flow was deter-
mined to be 5x10−12 kg/s and η ≈ 3.5x10−6 Pa·s, which leads to a Reynolds
number of Re = 47. The Reynolds number for the liquid helium flow exper-
iments in the smaller nanopore of 15 nm is Re = 2.8. Finally, laminar flow
is usually expected to be a good approximation for Reynolds numbers below
∼1800. Above this point, turbulence is expected to emerge and the dynamics
of the viscous fluid become fundamentally different.
3.3 Liquid Helium Flow
One of the most striking characteristic of helium at low temperature is a second
order phase transition at 2.17 K, below which the unusual properties of the
liquid prompted Kapitza [57] to name this new state of matter “superfluid”.
Above the transition, helium behaves like a normal liquid and its transport
properties are characterized by a well defined viscosity. Measurements of vis-
cosity fall into two types of experiments. Either flow is induced in a channel by
the application of a pressure head, or the moment of inertia of an oscillating
disk is measured. Classical liquids yield the same viscosity with both methods.
By contrast, the same experiments conducted with liquid below the superfluid
42 Theory of Mass Flow in Short Pipes
transition yield strikingly different results. Channel flow measurements of He-
II indicate a sharp decrease of the dynamic viscosity as temperature is lowered
below Tλ. Meanwhile, the oscillating disks experiments demonstrate He-II
possesses a substantial viscosity causing damping of the disk motion. These
apparently contradictory results have however been explained successfully by
the development of the well-known “two-fluid model”. Interesting historical
accounts of the discovery and characterization of the properties of the super-
fluid state of matter are readily available e.g. [58, 59] and only relevant results
will be presented here.
3.3.1 Two-Fluid Model
In the two-fluid model, the behavior of helium is explained by a dual descrip-
tion of complementary components that are viewed as simultaneously mixed
together, but with independent motion. The first component, the normal
phase, has physical properties matching the liquid helium of the He-I phase.
It also carries entropy and is responsible for the measurable viscosity. The sec-
ond component, the superfluid, is assumed to be a viscousless fluid undergoing
potential flow [60], i.e. a flow that is curl free and carries no entropy. The
normal component and the superfluid component are related through a flux
equation:
~Jtotal = ρs~vs + ρn~vn, (3.12)
where ρs and ρn are, respectively, the densities of the superfluid and normal
components of He-II, and are related to the total fluid density ρ = ρs + ρn.
3.3 Liquid Helium Flow 43
The fraction of superfluid composing the He-II phase approaches unity asymp-
totically as the temperature is reduced from Tλ to T= 0 K. A typical rule of
thumb is that He-II is essentially all superfluid below ∼1 K and the normal
component has a functional dependence on temperature close to
ρnρ
=
(T
Tλ
)5.6
up to Tλ. Figure 3.4 shows the calculated [61] fraction of normal density over
total density for bulk liquid helium.
Fig. 3.4 Fraction of normal density over total density of heliumin the He-II phase. The sharp decrease of the normal component isobserved at the superfluid transition and almost no normal heliumremains below 1 K. Data from Brooks [61].
3.3.2 Excitations in Superfluid Helium
While the two-fluid model was initially understood in macroscopic terms, Lan-
dau took a different approach and attempted to describe the superfluid state
44 Theory of Mass Flow in Short Pipes
with respect to its excitations. This microscopic theory of the He II phase was
designed semi-empirically to fit experimental data and explain the seemingly
intractable characteristics of this unconventional state of matter. Specifically,
the low temperature dependence of the specific heat followed a T3 power law
for T . 0.6, and nearly doubled between T = 1 K and Tλ. Such observations,
and Landau’s strong physical intuition, led him to posit the dispersion relation
of figure 3.5.
Fig. 3.5 Dispersion relation of the excitations in He-II. The lowestenergy excitations are characterized by a linear dispersion of smallwave-number. These collective modes are similar to longitudinalphonons in solids. Present at larger momentum p0 is the “roton”minimum, a collective excitation mode with an energy gap ∆. Theexcitations described by the convex dispersion relation at p0 areinterpreted as quantized rotational motion of small groups of Heatoms. The slope of the dashed and dotted lines give an estimateof the propagation velocity of each dissipation mode. Figure basedon figure 1.10 of [58].
At the lowest energy, the dispersion relation E(p) = pvph = ~kvph has a
linear behavior characterized by a large propagation velocity (vph ∼ 240 m/s).
3.3 Liquid Helium Flow 45
These collective modes are extended longitudinal waves similar to the phonon
modes in solids. The local minimum centered at p0/~ describes localized ex-
citations called “rotons”. Landau envisioned them as small groups of atoms
with a rotational motion and carrying an energy
E(p) = ∆ +(p− p0)2
2µ,
where µ is an effective mass and a gap ∆ is a thermal barrier limiting the emer-
gence of this dissipation mode. Neutron scattering experiments have confirmed
the qualitative shape of this dispersion relation for He II and the free param-
eters p0, ∆ and µ have been measured as p0/~ = 19.1 nm−1, ∆/kB = 8.65 K
and µ = 0.16m4, with m4 the atomic mass of helium.
Landau defined a velocity vL below which superfluid could flow without
dissipation. For larger velocities, excitations would be produced and the fluid
would dissipate energy. This is known as the Landau criteria for superfluidity
and, shown by a dotted line on the dispersion relation of figure 3.5, is given by
vL =
[E(~p)
|~p|
]min
,
where vL ≈ 58 m/s in bulk He II. Phenomenologically, the fact that the vL is
not zero in the dispersion relation of figure 3.5 allows the superfluid to propa-
gate without friction up to a critical velocity. All superfluid flow experiments
to date however observe a much smaller critical velocity, as seen in figure 3.6.
Further refinements were needed to characterize the dissipation in superfluids.
46 Theory of Mass Flow in Short Pipes
3.3.3 Vortices and Critical Velocity
Feynman suggested a dissipation mechanism where vortices traveling through
the superfluid could allow this potential flow to shed energy [62]. He first
assumed He II could be described by a macroscopic wavefunction Ψ = Ψ0eiφ,
where the amplitude Ψ = Ψ(~r, t) and phase φ = φ(~r, t) are defined at all
positions ~r and over time t. The amplitude of the wavefunction is normalized
as |Ψ|2 = ρs/m4, for particles with a mass of four atomic mass units. The
macroscopic phase φ is assumed to vary slowly and is related to the superfluid
velocity as
~vs =~m4
~∇φ.
A consequence of the macroscopic phase of the quantum fluid being unique
at all positions is that the circulation Γ around a closed loop must be quantized,
Γ =
∮~vsd~l = nκ4, for an integer n,
where κ4 = 2π~/m4 is the quantum of circulation. A quantized circulation
also has a secondary consequence on the formation of vortices. In the central
region, where the constant-phase lines would meet, the wavefunction must
vanish in order for the conditions of the superfluid potential flow to be met.
This region, where the Ψ (and thus ρs) drops to zero, is labeled the vortex
core, and has a size a0. Feynman finally argued that the creation of such a
vortex within the superfluid would be energetically favorable when the velocity
reaches
3.3 Liquid Helium Flow 47
vF ≡κ4
2πDln
(D
a0
), (3.13)
where D is the channel diameter. The core a0 is approximately 2 A and grows
with temperature up to the superfluid transition temperature. In this picture,
the vortex can continuously funnel energy away from the superfluid and keep
the flow at a fixed velocity[63]. Interestingly, the critical velocities predicted
with this new mechanism are quite close to the experimentally measured ones.
Indeed, most experimental critical velocities in larger capillaries (> O(10−6)
m) were indeed found to follow roughly the “1/D” size-dependence of equation
3.13 (see figure 3.6).
The agreement between experimental critical velocities and the Feynman
dissipation model contributed to the rapid acceptance of quantized vortices as
an alternative to Landau’s roton modes. Some experiments eventually found,
however, that in smaller pores, the critical velocities could be much larger,
independent of channel size and have a marked temperature dependence [65,
66, 67, 68]. Some of these results are shown in figure 3.6 with the symbol.
Importantly, those experiments found a linear temperature-dependence given
by
vc(T ) = vc0
(1− T
T0
). (3.14)
In this equation, vc0 is the critical velocity extrapolated to T = 0 K and T0 is
a fit parameter typically between 2.2 K and 2.8 K.
To further understand the dissipation mechanism behind these new results,
48 Theory of Mass Flow in Short Pipes
Fig. 3.6 Critical velocity measured in several experiments overthe last few decades. Circles () are experiments where a tempera-ture dependence was observed. Other symbols represent results ofexperiments where the critical velocity is temperature-independentbut still varies with channel size. The solid line is the Feynmancritical velocity of equation 3.13. Points in the nanometer regionare from experiments in helium film flow and the smallest systemspreviously accessible. Reproduced from [64].
very sensitive experiments were designed to measure superfluid flow in micro-
capillaries [69, 70, 71]. The results of these experiments are that vortices
are nucleated near the channel and the flow effectively experiences “phase-
slips”. This spontaneously reduces the superfluid velocity and leads to a critical
velocity. The nucleation of such vortices was found to be thermally activated
following an Ahrhenius-like form,
3.3 Liquid Helium Flow 49
Γ =ω
2πe− EakBT , (3.15)
where ω is the attempt frequency of the nucleation and Ea is the activation
energy, i.e. the energy that must be surpassed by the thermal fluctuations.
The specifics of this stochastic nucleation process have been quantified by
several authors and thorough reviews of the subject are available [64, 72]. For
the purpose of this thesis, the temperature and size dependence of the critical
velocity are of particular interest, since they offer important clues about the
nature of dissipation in superfluids.
50
51
Chapter 4
Sample Preparation and Flow
Measurement Technical Details
The development of controlled fabrication of very small pores [73, 74] has been
largely driven by research on bio-molecule separation [75, 76, 77], where dis-
crimination of single molecules can be achieved in channel diameters having
roughly the same size as the molecules studied. Many techniques have since
been developed for fabrication of tailored nanopores with virtually any diam-
eter [78, 79, 80].
In addition to fabrication techniques involving chemical etching [81] or
biological structures [82], one strategy has been to use a focused beam of en-
ergetic charged particles to gradually remove material in a thin membrane to
effectively drill a single aperture. This controlled ablation can be achieved
with various ionic species, but the best precision is obtained using the elec-
tron beams of field-effect transmission electron microscopes (TEM). This re-
52 Sample Preparation and Flow Measurement Technical Details
cent development has allowed, for the first time, the controlled production
of nanostructures down to less than one nanometer. These approaches for
tailored fabrication of nanopores can be compared to the intuitive option of
simply using pre-formed nanochannels, such as carbon nanotubes. While these
nanotubes have the desired small cross-sectional opening, they are typically
produced in large quantity and are thus available as membranes composed of
large aggregates of aligned nanotubes[83]. These are therefore not suited for
our experiments.
4.1 Sample Preparation
4.1.1 Solid-State Silicon Nitride Membranes
Recent advances in the electronic imaging industry have made it possible to
manipulate electron beams with exquisite control. Such precise control allows
one to focus the beam on a surface and expose only a few square nanometers.
This is the chief method that we have chosen in order to fabricate nano-
channels with the required length scale.
As mentioned above, our nanopores were made by electron-beam drilling
in a very thin membrane. These free-standing membranes are commercially
available, and many companies manufacture them. Unless otherwise noted,
all samples used in this thesis were produced using wafers from Silson Inc. R©.
They were manufactured by low-pressure chemical vapor deposition such that
a uniform layer of amorphous Si3N4 was grown over the whole silicon surface.
These membranes were then formed by chemically etching the silicon within
a small area of the 200 µm thick wafer pre-coated with the silicon nitride
4.1 Sample Preparation 53
Fig. 4.1 Photograph of a typical wafer used to fabricate nanoporesamples. In the center of the wafer, a pyramidal pit can be seenand is the result of the etching of an area of silicon. This selectiveetching leaves only the free-standing silicon nitride membrane (onthe bottom face here).
layers. This etching only removes the silicon and thus leaves a bare, free-
standing Si3N4 film, covering an empty pyramidal pit, as shown in figure 4.1.
The silicon wafers are typically square shaped (2x2 mm) and the free-standing
Si3N4 membrane are squares 30 to 50 µm wide.
The experiments presented in this work were conducted using membranes
with a thickness of 30, 50, 75 or 100 nm. Since nanopores are drilled through
the membranes, their thickness de facto defines the length of the nanopores.
When the length of nanopores is not critical to a given experiment, thinner
membranes are usually preferred because they allow much faster nanopore
drilling, which reduces the risk of producing irregular-shaped nanopores. The
trade-off of using thin membranes is that their low fracture strength limits
the maximum differential pressure they can withstand. We have empirically
determined that the limiting pressures (i.e. membrane-breaking pressure) were
∼45 bar for 50 nm thick membranes, and 6.7 bar for 30 nm thick ones. If larger
pressure differentials are required in future experiments, stronger membranes
54 Sample Preparation and Flow Measurement Technical Details
could easily be produced by etching a smaller area of the wafer, effectively
decreasing the surface area of free-standing silicon nitride.
The thickness of the membrane has a large impact on the time scale re-
quired to ablate and pierce through the SiN. In particular, if the sample stage
inside the microscope has no drift and the electron beam stays in one area, a
30 nm membrane will typically be pierced in a few seconds, whereas it would
take 60 seconds for a 50 nm one and up to several minutes for membranes
100 nm thick.
4.1.2 Sample Fabrication
We fabricated the nanopores using the field-effect transmission electron micro-
scope JEM-2100F located at the Ecole polytechnique de Montreal. Inside the
TEM, the electrons are accelerated by voltages up to 200 kV and are controlled
with magnetic lenses. A careful optimization of the beam intensity and focus
allows us to achieve a beam spot size of barely a few nanometers wide. If this
electron bombardment is maintained on the same area for several seconds or
minutes, enough material is removed to form a nanopore in the membrane.
Once this initial pore is opened, it is possible to move the sample stage a
few nanometers away so that the beam becomes aligned with the edges of the
nanopore. This forces the atoms from the pore wall to be removed gradually
in a controllable manner. With this technique, the nanopore can be carved out
into a cylindrical shape and up to any diameter, from 0 to more than 100 nm.
We have fabricated hundreds of such pores during the process of preparing
samples for the experiments presented in this thesis and in figure 4.2, I show
4.1 Sample Preparation 55
a few typical samples.
Finally, I also point out that this technique has a big advantage in that
it allows one to “see” the nanopore while it is being drilled. A picture can
therefore be taken immediately after fabrication. This direct imaging thus
makes it possible to readily determine the dimensions of the nanopore and
discriminate samples based on the requirements of each experiment.
Nanopore Geometry
In principle, the precision of the electron beam gives great control over the
effective hole size and allows a very careful fabrication of nanopores [84]. While
true, there are however practical difficulties in using magnetic lenses to control
a beam of charged particles to carve out a hole in an amorphous material like
silicon nitride. In particular, slight distortions in the beam’s cross sectional
profile may affect the resulting nanopore and produce a non-circular cross
section; in other words non-gaussian beams can produce irregular drilling.
This is especially relevant since the drilling is conducted at high magnification,
whereas the pore is imaged at lower magnification. This means that the pore
seen during the drilling can appear perfectly circular at high magnification, but
upon further imaging at low magnification, the actual shape is usually found to
differ slightly. Because of this risk of distortion of the nanopore shape during
the drilling procedure, careful images of each nanopores were taken after each
fabrication, at a magnification low enough to avoid distortions. Finally, the
calibration of these images allowed us to determine the nanopore sizes directly
after fabrication.
56 Sample Preparation and Flow Measurement Technical Details
Fig. 4.2 Transmission electron pictures of different nanoporesranging in size from 300 nm to less than 2 nm in diameter. Thenanopores were fabricated by electron ablation of a silicon nitridemembrane suspended on a silicon wafer. The white bars are respec-tively 50 nm for pictures A-D and 10 nm for pictures E-H.
4.1 Sample Preparation 57
Regarding the nanopore profile, it is generally assumed that the ablation
of material is initiated simultaneously from both sides of the membrane and
continues until the two aligned cavities connect halfway to form a fully opened
channel. If one considers the beam cross sectional intensity to be decreasing
radially, then the outer edges of the electron beam are less dense and so will
not drill as fast as in the center of the beam. This means that the ablation
is more efficient along the axis of the electron beam, but also that the area
around the nanopores has potentially been bombarded as well, albeit at much
smaller intensity. Indeed, 3D tomography analysis of the nanopores [85] shows
that the ablation of material with highly energetic electrons does not produce
a perfectly cylindrical pore, but rather an “hourglass” shaped one. Despite
this, it is possible to alter this initial pore shape with a secondary treatment
that straightens the pore walls. This treatment consists of an exposure of
the area surrounding the nanopore to a low-intensity beam for a short time
[86]. It effectively gives the atoms near the nanopore edge enough energy to
redistribute themselves and “flow-back” in the pore. In fact, we have observed
repeatedly that this technique has helped stabilize the SiN surface and reduces
nanopore deformation over time.
Interestingly, this same method can also be used to shrink the nanopores.
In fact, a longer exposure time under low intensity can maintain the fluidization
of nearby atoms and keep filling the nanopore until it eventually closes [87, 88].
We have used this method to produce nanopores smaller than 3 nanometers,
as can be seen in figure 4.2G,H. The parameters for this procedure are however
not always reliable and more work is required to fully capture the subtleties
58 Sample Preparation and Flow Measurement Technical Details
of the physics at play during the shaping down of solid-state nanopores.
Nanopore Opening Angle
The mass transport experiments described in this thesis are very sensitive to
the diameter and opening angle of the nanopore. In essence, the conductance of
a pore is intrinsically determined by its geometry and it is therefore important
to determine, as much as possible, the three-dimensional shape and profile of
pores.
The intensity of the TEM picture can be used to provide an estimate of
the shape of the nanopores. To first order, the intensity of the transmitted
electrons is proportional to the thickness of SiN. Thus, making use of the
average saturation of a group of pixels, a “thickness” can be estimated. We
took the average of pixels in the center of the pore as an intensity background
to define a thickness of zero. We can do the same for the complete thickness
using the pixels away from the pore. This way, a calibration is obtained as a
function of the saturation in each point of the picture. If we further assume
a symmetric material ablation with respect to the middle plane of the wafer
[85], we can reconstruct a 2D profile from the images. In order for the profile
to be clear, the high amount of fluctuation in intensity in the TEM picture
must be averaged over adjacent pixels. Grey points in figure 4.3B show the
equivalent thickness of all pixels without averaging, whereas the black dots
show the thickness computed from a local average of pixel intensities.
The power of this relatively simple procedure is to gain information on
the shape of the nanopores that ultimately will be used in the mass transport
4.1 Sample Preparation 59
Fig. 4.3 TEM images (A) can be used to infer the thickness ofmaterial (B) the electrons go through. The intensity away from thepore is set as the thickness of membrane chosen at manufacturing,(in this case 50 nm), and the intensity in the center of the pore tozero. The intensities in between are normalized to this thicknessand we can infer the profile of the nanopore. This profile showsthe slight opening angle of ∼10 with respect the a perfect verticalchannel wall.
experiments. Since the true nanopore shape is not perfectly cylindrical, a
modified Clausing factor will have to be used to improve the modeling of the
data. Judging from nanopore profiles similar to that shown in figure 4.3,
it is reasonable to estimate the opening angle with respect to the vertical
axis at ∼10. For that sample, a quantitative comparison with the Knudsen
conductance of small pores will be shown in the next chapter.
60 Sample Preparation and Flow Measurement Technical Details
Nanopore Diagnostics with Electron Microscope
The difference between a partially obstructed nanopore and a fully drilled one
can easily be observed in the TEM images. Specifically, the spatial fluctuation
in the intensity measured by the CCD camera of the TEM are qualitatively
different when the electrons go through either vacuum or the membrane. This
means that a visual inspection of the TEM image is sufficient to determine
whether the pores remain open or close during storage or post-experiment.
Another feature of clean nanopores is the brighter boundary at the perime-
ter of the nanopore caused by electron diffraction off of the solid pore wall.
Figure 4.2A shows this clearly where the low-intensity noise in the center of
the nanopore contrasts greatly with the larger intensity fluctuations caused
by electron interference in the amorphous SiN. These characteristics in the
nanopore images show the focus of the electron beam is optimized and that
the surface is free of contaminants. When pores are blocked by contaminants,
the electron diffraction and transmission intensity are both affected in a way
that the user can readily identify.
Nanopores are at risk of becoming partially or fully blocked by contami-
nants at any time and the TEM can be used as a diagnostic tool to identify the
nature of this contamination. For example, we have used the energy dispersive
X-ray spectroscopy (EDX) capabilities of the TEM to determine the nature of
the material covering blocked pores. A strong carbon signature was a typical
result of this procedure and indicated external contaminants. In these cases,
the nanopores could not be redrilled and the whole wafer was assumed to be
contaminated, and was therefore discarded.
4.2 Flow Measurements Via Mass Spectrometry 61
4.2 Flow Measurements Via Mass Spectrometry
The fabrication technique described in the previous section is very advanta-
geous to create a single nanometric channel inside a solid-state membrane.
Historically, the simplest way to measure mass transport was to detect the
change in a thermodynamic variable of the fluid within the reservoirs on either
side of the transport channel [89]. For example, this could be the pressure
change in a fixed volume as some of the fluid inside flows out. This tech-
nique is, however, only effective for macroscopic apertures since, in very small
ones, the change in pressure becomes smaller than the precision of pressure
gauges, due to the large relative size of the source and drain reservoirs. In
other words, enough particles must flow out for the pressure change to be
discernible. A quick estimate shows that in a typical experimental enclosure
of volume O(10−5) m3 contains approximately 3·1020 gas particles at stan-
dard temperature and pressure. Typical mass flows in small apertures of size
100 nm or less are, at most, of the order of O(1014) atoms/s and, consequently,
a 1 % drop would take O(106) seconds, which is experimentally prohibitive.
A more efficient method to measure mass transport is to literally count the
atoms going through the nanochannel. One way this can be accomplished is
by pumping continuously on the drain reservoir below the pore and use a mass
spectrometer to analyze the gas flow. In particular, the mass spectrometer
inside a helium leak detector is optimized for accurate measurement of helium
flows. A similar method that has been shown to work in microchannels is to
use a residual gas analyzer [90].
In essence, the underlying measurements of the results I present over the
62 Sample Preparation and Flow Measurement Technical Details
next few chapters can be described as an accurate determination of a carefully-
tailored single leak. The leak detector I used to make all flow measurements
was a Pfeiffer Smart Test HLT560, with twelve orders of magnitude range
of sensitivity to flow rates. The Smart Test device also has the option to
discriminate for the mass of atoms sent to the detectors from 2 to 4 atomic
mass units. This means the lighter helium isotope 3He could be substituted
in the experiments and be measured in exactly the same way, although it has
not yet been attempted.
Mass spectrometers are designed to discriminate charged particles by their
mass. For neutral gas atoms, the first step is to pass them through an ioniza-
tion chamber where filaments emit a stream of electrons that ionize the atoms
so they can be accelerated by an electric field. This acceleration voltage V in-
creases the kinetic energy of each atom proportionally to its charge q, so that
qV = m|~v|2/2. The atoms are then passed through a homogeneous magnetic
field where they are deflected by the Lorentz force FB = q~v × ~B, which is
proportional to the speed |~v| of the atoms and the applied magnetic field B.
This perpendicular force induces a curved trajectory, with FB = mv2/r, such
that the radius r of each atom’s path is
r =mv2
qB=
1
B
√2mV
q,
where the speed v was rewritten with the voltage acceleration equation from
above.
Figure 4.4 shows a schematic drawing of a mass spectrometer. The three
curved paths represent the paths atoms or molecules of different mass would
4.2 Flow Measurements Via Mass Spectrometry 63
Fig. 4.4 Schematic diagram of a mass spectrometer. The gas issent through an ionization chamber where it becomes ionized andis accelerated by an electric field towards a region where a magneticfield deflects each ionized atom. The curvature of the path theseatoms take is dependent on their mass and a discrimination of theatomic species is achieved by blocking the paths of unwanted atoms,letting only the proper ones through to the detector.
follow when traveling through the magnetic field. The position of a slit can
discriminate which ions reach the detector. The ion detector then returns a
voltage proportional to the number of atoms with the chosen mass. A cal-
ibration of this signal was done regularly over the course of experiments to
ensure that the measured flow rate was quantitatively accurate. To do so, we
used a calibrated leak certified by the US National Institute of Standards and
Technology with a value of 2.79·10−8± 10% mbar·l/s. The flow rate of helium
gas through a pore can thus be accurately measured if the drain reservoir of
the experimental cell is pumped on by the leak detector.
64 Sample Preparation and Flow Measurement Technical Details
4.3 Measurement Scheme
In the previous two sections I have described the techniques necessary to fab-
ricate nanopores, as well as to measure the mass flows of gas particles in them.
I will now describe how to use these two elements to inform the design of an
experimental scheme suited for measuring the mass transport inside a single
nanopore. The strategy that we have used has great similarities with the mea-
surement of charge transport in simple electric circuits. This is because the
physics of electric current propagating in materials is very often described by
simple concepts of resistance to flow of an ensemble of electric charges. The
description of mass transport of electrically neutral particles can also be de-
scribed with very similar concepts if we make a direct analogy between charges
and mass, as well as electric potentials and pressure differentials. The charge
transport in a material with linear resistance R and under a voltage V is de-
scribed in its simplest form by Ohm’s law I = (1/R)V = G·V , where G ≡ R−1
is the electric conductance of the material. In the case of mass transport in
the linear regime, a similar equation holds,
Qm = G∆P,
where the mass flow Qm (in kg/s) is proportional to the pressure difference ∆P
between the ends of the conduction channel. The constant of proportionality is
the conductance G, defined analogously to the electrical conductance (inverse
resistance) in Ohm’s Law. The conductance of a nanopore is thus the main
characteristic that we will study in our exploration of the physics of flow within
4.3 Measurement Scheme 65
confined nanochannels.
4.3.1 Experimental Cell
The simplest strategy to detect mass flow in a single channel can be inspired by
the electrical circuit analogy. A pressure differential must be applied between
a source and a drain reservoirs positioned on either side of the conduction
channel. This pressure differential induces a flow through the channel. In this
design, it is critical that the particles in the upper reservoir, the source, can
only go to the lower reservoir, the drain, via the nanosized channel, otherwise
a secondary transport path would potentially dominate the signal. This is
because the conductances of parallel transport paths are summed in transport
measurements. In other words, the wafer holding the nanopore of interest
must be perfectly sealed between the source and drain reservoirs.
Many versions of an experimental cell were designed and built throughout
the course of this work, but only two are presented here. One key constraint
limiting the design is that many samples should be easily substituted in the
same cell without requiring too many repairs. This rules out, for example,
using epoxy to seal a membrane directly inside an experimental cell. The
whole cell should also remain leak-tight at all temperatures below that of
the laboratory, and hold for internal pressures from approximately a hundred
atmospheres down to vacuum. Other than permanent solutions like welding
or viton R© o-rings that lose elasticity at low temperatures, the best strategy to
adopt here is to use a compressible wire made of soft metal, such as indium,
as a non-permanent sealant. Indium wires with diameters smaller than ∼700
66 Sample Preparation and Flow Measurement Technical Details
µm can be compressed between two flat surfaces and make a joint that will
withstand multiple temperature cycles from room temperatures to cryogenic
temperatures and remain leak-tight.
First Generation Experimental Cell
The first generation of experimental cells that we built were similar to the
one shown in figure 4.5 and the photograph in figure 4.9B. The experimental
cell is composed of three main parts: a top flange, a membrane support in the
middle (piece T on figure 4.5), and the main cell body. The membrane support
piece is easily removable and many copies were fabricated to allow for multiple
samples to be prepared in parallel and thus allow faster turnaround between
experiments. The body of the cell is made from high thermal conductivity coin
silver (10%Cu,90%Ag) to ensure the best possible thermal equilibrium between
the cell and the helium fluid inside it. The membrane support piece is made of
Invar 42 R©(42%Ni) because of its very low thermal contraction coefficient that
closely matches that of the Si wafer. Stainless steel capillaries are soldered
to the top flange and to the body of the cell to bring helium in and out,
respectively. These capillaries are then connected to a gas handling system
that will be described in section 4.3.2.
A ∼1 mm diameter indium o-ring compressed between the top flange and
the cell body ensures that the inside of the cell is completely sealed from the
outside. A similar seal, albeit smaller, is also made between the membrane
support and the cell body to prevent helium above the sample from reaching
the bottom volume. Otherwise, this would generate an unwanted secondary
4.3 Measurement Scheme 67
T
S
M
D
Fig. 4.5 Three-dimensional model of an experimental cell de-signed for mass transport measurements on a nanopore (purplewafer in the middle of the model). A top flange (brass colored)and a silver body are sealed together enclosing the wafer with thenanopore of interest. The upper reservoir, source (S), is separatedfrom the lower reservoir, drain (D), by the leak-tight membrane (M)such that transfer of gas can only happen through the nanopore.The wafer is mounted on a support piece (T) that is screwed andsealed to the body of the cell (silver colored). Pressure in the source(inlet) reservoir can be increased while the drain (outlet) reservoiris kept under vacuum. The capillaries leading in and out of theexperimental cell are connected to a gas handling system. The flatbottom of the cell can be securely fastened to a metallic surface toensure a strong thermal link.
flow channel. The space between the Si wafer and the Invar support is the last
location to seal in order for the top and bottom reservoirs to be unconnected.
This seal is prepared in advance and is achieved using either one of two epoxies:
Stycast 2850 or Armstrong A-12 epoxy.
The use of epoxy has a major disadvantage compared to an all-metal joint:
helium can diffuse through it. This diffusion is however not detectable at liquid
nitrogen temperatures, yet it can be as high as 10−16 kg/s at room tempera-
68 Sample Preparation and Flow Measurement Technical Details
ture. Importantly, it is not possible to reliably correct for this diffusion signal
because it is not reproducible and fluctuates in time. All data presented in
the thesis is therefore either taken at 77 K, or for mass flows several orders of
magnitude larger than the diffusion in the case of room temperature measure-
ments. A second disadvantage of epoxies is that they are at risk of producing
volatile compounds that can contaminate the surface of the sample and possi-
bly cover the nanopore. Finally, epoxies need to be cured for extended periods,
typically hours, which limits the rapidity of sample preparation. These issues
with epoxies have influenced the design of an all-metal experimental cell.
Second Generation Experimental Cell
The second generation of experimental cell was developed with the goal of re-
moving all non-metallic content from the experimental cell. Figure 4.6 shows
a cross section of the central region of the second generation experimental cells
used to take measurements of mass flow in nanopores. This design is charac-
terized by the removal of the support piece and the use of an indium ring seal
located directly under the silicon wafer. A thin Invar disk is used to apply even
compression on the wafer, which then compresses the indium underneath it.
Once compressed, the indium wire can remain leak-tight over any temperature
range attained in our experiments and has the added advantage of requiring
only a few minutes of preparation. Additionally, the removal of a compressed
indium wire is much simpler than for glues or epoxy sealants since it can be
pealed off the surface by applying a gentle upward force.
Both the silver and Invar surfaces touching the wafer are milled flat to
4.3 Measurement Scheme 69
avoid damaging the fragile silicon; rough surfaces can easily lead to cleavage of
the whole wafer when some compression is applied. Care must be taken when
aligning the free-standing window with the hole in the silver bottom part since
the SiN membrane can withstand virtually no physical contact.
Brass
Invar
Ag
In
Si
Fig. 4.6 Schematics of a cross section of the central region of thesecond generation of experimental cells. The top flange and bodyare made of coin silver (Ag90%Cu10%), the silicon wafer is compress-ing an indium o-ring to provide the seal between top and bottomreservoirs and an INVAR 42 disk is used to clamp the wafer. All di-mensions are to scale except for the pit of etched silicon. The screwsare made of a brass alloy that will contract slightly at lower tem-peratures to ensure the indium seal remains tight during thermalcycles. A second indium seal (outside of visible area), compressedbetween the coin silver top and bottom flanges, is used to seal theinside helium from a leak to the outside environment.
The volume above the membrane is designed to be large in order for the
helium fluid to thermalize properly with the silver surfaces. A thermometer
and a heating coil can be secured directly to the silver parts on the outside of
the cell. The helium inlet and oulet capillaries are connected to the cell using
adapters screwed on either side of the cell with indium o-rings joints. For low
temperature experiments, the inlet capillaries go through multiple stages of
heat thermalization as will be explained in section 4.4.
70 Sample Preparation and Flow Measurement Technical Details
4.3.2 Gas Handling System
The flow of helium through the nanopore is created by the difference in pres-
sure between the top (source reservoir) and bottom (drain reservoir) of the
cell. A precise control of the pressure applied above the sample is therefore
very important, and this is achieved using a gas handling system (GHS). Fig-
ure 4.7 shows a photograph of the GHS. As can be seen from the photograph,
multiple pressure gauges are required at specific points in the GHS to monitor
processes such as filtration, storage or pressure increase/decrease. All valves
and capillaries in the GHS are made of stainless steel with all-metal joints, and
all are designed to withstand internal pressure up to 15000 psi (∼1000 atm).
The capillaries have an internal diameter of 0.000761 m, which is small enough
to ensure that the total volume in the capillaries does not dwarf the experi-
mental cell volume during experiments, but still large enough to have a low
impedance to flow compared to the nanopore.
Using helium as the test gas for mass transport experiments has the great
advantage that it can be passed through filters at liquid nitrogen tempera-
tures and fine-meshed molecular sieves. Five such filters were used to prevent
contamination from air introduced in the cell during sample preparation as
well as filter the gas during transfers between components of the GHS. Two of
the filters are canisters filled with activated charcoal granules that have a very
large surface area because of their natural porosity. The adsorption potential
of the surfaces is highly dependent on temperature and keeping this filter im-
mersed in a bath at 77 K can help trap contaminants while letting helium gas
through. As long as the filter is kept cold, the molecules will remain trapped.
4.3 Measurement Scheme 71
While helium is conveniently not stopped by the filters, it will be adsorbed
by activated charcoal if the temperature is reduced further. This is the prin-
ciple used in the operation of a helium gas dipstick. A small vessel filled with
charcoal is designed such that it can be transferred between a liquid helium
bath and a liquid nitrogen one. In the colder environment, the charcoal sur-
faces adsorb much more helium gas and the pressure in the vessel decreases.
Two such dipsticks are part of the GHS and if the valve leading to them is
opened, the decrease in pressure can also be used to pump sections of the GHS.
The second important use of the dipsticks is to access large pressures. This
pressurization can be achieved once a large quantity of helium gas has been
accumulated in the charcoal and the dipstick vessel is returned to the liquid
nitrogen bath. The higher temperature induces an instantaneous desorption
that creates a pressure build-up. A large gas tank of approximately 0.003 m3,
seen at the bottom of figure 4.7, is filled to an internal pressure of several
atmospheres and contains sufficient gas to pressurize the experimental cell
well above 100 atm using the thermal cycling of the charcoal dipsticks. This
pressurized helium can then be directed towards the experimental cell in order
to change the applied pressure on the nanopore. The cycle can be repeated by
returning the dipstick to the liquid helium bath and the pressure in the cell
can thus be decreased. The whole operation remains in a closed loop cycle.
4.3.3 Electrical Circuit Analogy
As already mentioned, a pressure differential can create a mass current through
nanopores. Figure 4.8 shows a simplified representation of the experimental
72 Sample Preparation and Flow Measurement Technical Details
Fig. 4.7 Photograph of a portion of the gas handling system usedto control the helium gas. Visible near the top are several pressuregauge displays, high pressure valves connected by stainless steelcapillaries and the icons showing the location of other filters, gaugesand the experimental cell. A storage tank of ∼3 liters sits at thebottom of the panel. Not shown are the digital pressure gauge, theliquid nitrogen dewar and pressurization dipsticks.
setup to measure a mass current through a linear system like the one proposed
here, as well as its electrical circuit equivalent. Figure 4.8A shows an exploded
view of the experimental cell with the conductance of the capillaries explicitly
4.3 Measurement Scheme 73
indicated. This finite conductance is in practice positioned in series with the
conductance of the nanopore and it should therefore be estimated.
Fig. 4.8 (A) Exploded view of the experimental cell with capil-laries leading in and out having respective conductances GS andGD. (B) Cartoon diagram (not to scale) of the whole experimen-tal circuit where a reservoir on the left contains pressurized heliumtraveling in a capillary with a very high conductance estimated us-ing hydrodynamics laminar flow theory. In the middle is a smallertube representing the nanopore and finally the tubes to the rightlead to the mass spectrometer (AI) that counts atoms. This lattersection of the circuit has a high conductance given by the theory ofdiffusion in tubes under vacuum. (C) An electric circuit equivalentwhere resistances are shown with proper value relative to one an-other. Electric current is measured in a similar way to the proposedmass current measurement.
From figure 4.8C, it is clear that the equivalent electrical circuit has many
resistances effectively in series and the total resistance of the system is thus the
sum of all resistances. The capillaries leading in and out of the experimental
cell must have sufficiently small resistances Rcap for the nanopore resistance
Rnp to be approximated by the total resistance Rtot. In other words,
Rtot =∑
Ri = Rn and Rnp Rcap.
74 Sample Preparation and Flow Measurement Technical Details
Conversely, the conductance of the nanopore must be significantly smaller than
the conductance of the other components in the GHS if the measured mass
flow is to be attributed only to the nanopore.
The mass flow measured at the mass spectrometer is determined by the
total conductance G−1tot = G−1
S +G−1D +G−1
np of the circuit, where GS,D are the
source and drain conductances. A calculation of the conductance of the cap-
illaries can be done using the infinite length approximation for Poiseuille flow
(equation 3.7) (GS ≈ 10−11 m·s at ∼1 bar) and Knudsen free-molecular diffu-
sion (GD ≈ 10−13 m·s at 10−3 mbar) respectively for the inlet and outlet cap-
illaries. In comparison, the typical nanopore conductance is Gnp ≈ 10−18 m·s.
A more intuitive perspective emerges by completing the analogy with the elec-
trical circuit. As shown in figure 4.8C, setting the nanopore resistance to 1 MΩ
and scaling the other elements properly, the resistance of the GHS capillaries
are found to be negligible. This comparison makes it clear that the total flow
rate measured by the mass spectrometer is attributed only to the nanopore
and the effects of the macroscopic capillaries can be neglected.
4.4 Cryogenics and Thermometry
Conducting gas flow experiments at low temperature has a few advantages
compared to experiments at room temperature. The first difference is in the
proclivity of cold surfaces to adsorb contaminants. In the same way that
the cold traps and filters operate, cold surfaces along the path of the helium
have an increased potential to capture some of the contaminants that might
be present within the gas. In fact, a corollary to this is that desorption off
4.4 Cryogenics and Thermometry 75
surfaces is reduced at lower temperatures. Keeping the experimental cell cold
can thus help reduce the risk that organic compounds become volatile and
block the nanopore. Contamination of the wafer surface has also been observed
several times and the nanopores have indeed been blocked on several occasions,
especially for smaller diameter nanopores. Most measurements presented in
this thesis were taken at temperatures equal to, or below the liquid nitrogen
boiling point. The second advantage of working at lower temperatures is the
larger signal, which is approximately a factor of two larger because of the√T
dependence of Knudsen effusion (equation 3.6) and the density dependence in
the viscous flow (equation 3.10). In fact, the next chapter will discuss this
temperature dependence in more detail.
The most important advantage of having access to low temperatures is the
ability to make clean transport measurements in the liquid state, and more
interestingly in the superfluid phase of 4He. To achieve liquefaction of helium
and observe the transition to superfluidity, temperatures below 2.17 K are
necessary, and this is conveniently well within the reach of a 3He cryostat.
Results from experiment using the 3He cryostat will be presented in chapter 6.
4.4.1 The 3He Cryostat
A cryostat consists of a multi-stage cooling apparatus that is inserted into a
large dewar designed to minimize transport of heat from the outside environ-
ment to the internal components. This dewar consists of several volumes kept
either under vacuum or filled with cryogenic liquids. These act as barriers to
heat conduction and help reduce the unavoidable heat leaks in the cryostat.
76 Sample Preparation and Flow Measurement Technical Details
Figure 4.9 presents a simple diagram of the different stages of the “insert”
of the 3He cryostat. The main cooling components of the cryostat are the
charcoal sorption pump, the 1K-pot, the 3He-pot and the sample stage. All
these are kept in a vacuum chamber, which is immersed in the liquid helium
bath. The core components of the cryostat were purchased from a commercial
supplier (Janis), whereas all experiment specific parts were manufactured and
designed by our group. The details of operation of the cryostat follow.
The 3He Cryostat Operation
The 3He cryostat uses a simple evaporative cooling technique to reduce the
temperature sequentially from the temperature of a helium bath at saturated
vapor pressure, to the temperature of a helium bath at low pressure, and then
to the final base temperature of a 3He bath at low pressure. The sample stage is
in thermal contact with the 3He bath in order to take advantage of the coldest
location. The base temperature is between 260 mK and 340 mK depending on
the thermal leaks caused by capillaries and conducting wires anchored above
at higher temperatures.
The first stage of the cooling procedure is to thermalize all the insert com-
ponents with the helium bath. This is achieved by introducing helium gas
inside the vacuum chamber such that heat from the insert is conducted to the
vacuum chamber walls that are in direct contact with the helium bath. Once
the temperature of all elements on the insert reaches 4.2 K, the exchange gas is
removed such that the vacuum chamber acts as insulation between the cham-
ber wall and the insert components. This is a necessary step in order to reach
4.4 Cryogenics and Thermometry 77
colder temperatures.
The next step is to let some liquid helium from the helium bath enter
the chamber called the “1 K pot”, through the needle valve seen on figure
4.9. If the vapor pressure of the liquid helium inside the 1 K stage is reduced
by pumping on it, the thermodynamic equilibrium shifts and the temperature
decreases as more helium evaporates to replace the atoms removed by pumping.
This situation can be stabilized by supplying warm liquid helium at ∼4 K to
balance the evaporated gas being pumped away. In this manner, a temperature
of ∼1 K can be maintained under optimal conditions. If the balance is slightly
shifted, for example by increasing the pumping rate of the gas or reducing
the influx of liquid, the overall temperature will decrease, albeit for a limited
period. This unbalanced situation would slowly reduce the ratio of liquid to
gas and once all the liquid is evaporated, the cooling would stop and the ever
present heat leaks would cause the temperature to increase sharply. A more
desirable solution for sustained operation is to keep a balanced pumping rate
versus liquid injection. This balance can be achieved in practice for extended
periods and temperature remains stable within a few millikelvins.
Cooling to base temperature is achieved in the 3He chamber with the same
evaporative cooling technique described above. The scarcity of 3He gas makes
it impractical to have a continuous supply of atoms and the design is therefore
made for closed-cycle operation. It would be too costly1 to pump away the 3He
gas. Only a few liters of gaseous 3He is therefore kept in a storage container
at room temperature. This storage container is welded to a tube that passes
1Current prices can be as high as ∼$8000 per liter of gaseous 3He at STP.
78 Sample Preparation and Flow Measurement Technical Details
Fig. 4.9 Diagram of the principal 3He cryostat components. (Inblack) The vacuum chamber immersed in the helium bath of theexternal dewar. (In teal) Charcoal sorption pump parts. The cen-tral tube leads to a storage vessel above (not visible) and, below,connects to the 3He pot (in green). (In red) The liquid helium inletletting liquid into the 1 K pot, passing through the needle valve. Onthe opposite side is a pumping line. (In grey) The coldest point ofthe insert and the volume dedicated for samples. On the right sideis a photograph of the insert matching the schematics on the left.The diagram is modified from CAD drawings by the manufacturers,Janis Research Company, LLC.
4.4 Cryogenics and Thermometry 79
through the center of the cryostat insert and ends in the 3He chamber where
the evaporative cooling can occur. As shown in figure 4.9, a charcoal-filled
chamber is also present in the middle of this closed system such that 3He
atoms are always free to move between the room temperature storage vessel,
the charcoal filled canister, and the 3He pot.
At this point in the cooldown procedure of the cryostat, all components
are at 4 K except the 1 K pot where evaporative cooling of 4He is taking place.
As explained in the case of the pressure building dipsticks, activated charcoal
becomes very efficient at adsorbing atoms when cooled down to liquid helium
temperatures. Having a chamber filled with charcoal directly connected to the
stored 3He from the storage container, which remains at room temperature at
all times, most of the 3He gas will get adsorbed by the charcoal. This means
that the charcoal chamber acts as a temperature-controlled pump and is the
reason why this component is also called a charcoal sorption pump.
A section of the tube leading from the charcoal sorption pump to the
3He pot passes through the 1 K pot such that some 3He gas can condense
on the inner walls of the tube if the 1 K pot temperature is reduced below
approximately 1.8 K. By raising the temperature of the charcoal chamber
above ∼37 K, it is possible to reduce the adsorption potential in a way that
3He gas will desorb and fill the whole closed system volume. The resulting
increased pressure induces liquefaction of some 3He atoms when coming in
contact with the tube region cooled by the 1 K pot. The gas will continue
to condense and fill the bottom reservoir with liquid 3He. It should be clear
that no mixing of the two isotopes occurs throughout this procedure, unlike
80 Sample Preparation and Flow Measurement Technical Details
in the case of the dilution refrigerator cryostats. Heat transfer is exclusively
by diffusion through the metallic structure.
Once all available atoms are condensed, the temperature of the charcoal
can be lowered by pumping on the liquid 3He. This will lower the temperature
of the 3He bath and the samples in thermal contact with it. The colder the
charcoal is, the more 3He can be adsorbed and the more pumping power is
available to increase evaporation of 3He atoms and reduce the temperature
of the remaining liquid in the 3He pot. In practice, this pumping-dependent
temperature control only offers a coarse modulation of the sample temperature,
and usually has to be supplemented with one or several heating elements that
can be fine-tuned to balance the power input (heating) to the cooling power
of the cryostat and regulate temperature with millikelvin temperatures. This
fine-tuning is achieved through a feedback loop between a thermometer and a
heating element.
Temperature Control
A Lakeshore temperature controller model 340 was used to control a PID loop
between a 25 Ω heater and a ruthenium oxide thermometer, both affixed to
the experimental cell. The P, I and D coefficients where adjusted manually in
each temperature range for efficient temperature modulation compared to the
set-points chosen.
Given that the cooling power of the cryostat at base temperature is on
the order of milliwatts, the 25 Ω resistor used for heating the sample can
easily raise the temperature sharply and the proportional (P) coefficient was
4.4 Cryogenics and Thermometry 81
therefore kept low (<5) to avoid uncontrolled warming that could potentially
induce boiling of the liquid helium inside the cell. This is especially relevant
in the superfluid transport experiments where temperature changes of a few
milli-Kelvin where often desired. The “I” coefficient was simply chosen to
avoid underdamped/overdamped oscillations of the temperature with respect
to the temperature set-point. The experiments I performed with the cryostat
never required the setpoints to depend on time (∂Tset/∂t = 0) so D = 0 was
the de facto value for the derivative component in all PID loop operation.
It is worth noting that the temperature control loop is only effective if the
temperature is measured accurately. This means a thermometer with high
precision must be used and the thermometer must be securely fastened to the
component of interest so that a possible temperature gradient does not affect
the accuracy of the reading. These two requirements are covered in the next
subsection.
4.4.2 Thermometry and Thermal Anchoring
The helium used in the experiments is stored in the GHS at room temperature,
so efficient cooling of the gas is necessary in order for transport measurements
to be realized at low temperatures and in the liquid phase. Reducing the
temperature of the gas can be achieved by letting it go through a volume
with cold surfaces, as was explained in section 4.3.2. The simplest option for
this cooling is to have the experimental cell submerged in a cryo-liquid, along
with a section of the capillary, such that gas from the inlet capillary is cooled
and thermalizes within the cell. This is the method that was used for most
82 Sample Preparation and Flow Measurement Technical Details
measurements in chapter 5 and was shown to be effective for temperatures at
and above the liquid nitrogen boiling point.
Fig. 4.10 (A) Thermometer inside an OFHC copper piece milledin a bobbin shape so that the electrical leads can be wound aroundthe post to ensure that the whole unit is well thermalized. Theruthenium oxide resistive element is (hidden from view) inside thebobbin within a protective canister. (B) Exploded view of a heatexchanger where the bottom part is filled with pressure-sinteredsilver powder and capillaries are soldered to the two posts. Theseposts are brazed to the top of the small container. Superfluid heliumentering from one capillary into the heat exchanger will rapidly fillthe silver powder’s crevices and be in thermal contact with thewhole unit.
The second option is to use multiple cooling stages called thermal anchors
(figure 4.10B). This method of cooling is required for experiments in the cryo-
stat where the energy from the warm gas could easily overwhelm the cooling
power of the 3He pot and prevent proper operation. The strategy is therefore
to force the gas from room temperature through increasingly cold thermal an-
chors to remove as much energy as possible before letting the gas thermalize
with the cell. Specifically, I used a thermal anchor on the vacuum can at 4.2
4.4 Cryogenics and Thermometry 83
K, another on the 1 K pot and a last one on the 3He pot. It should be noted
that the capillary from the outlet of the experimental cell was kept insulated
all the way to the vacuum can’s thermal anchor. This capillary was made
of an alloy of copper and nickel with a thermal conductivity on the order of
O(10)−4 W/cmK, on par with Teflon and manganin, and an order of magni-
tude lower than standard metals. Since this tube is ∼60 cm long, it does not
represent a serious heat leak to the bottom of the cell. As mentioned in section
4.3.1, we chose coin-silver as the material for the experimental cell because of
its large thermal conductivity, which allows the helium to quickly thermalize.
The cell’s thermal conductivity is also necessary to measure the temperature
with thermometers affixed to the body of the cell. It is a good practice to
securely fasten thermometers and heaters directly on the experimental cell to
minimize the risk of temperature gradients.
Several thermometers (figure 4.10A) are monitored continuously during op-
eration of the cryostat and all of them use a 4-point measurement technique
to avoid the effects of the change in lead resistance as a function of tempera-
ture. Both the sorption pump and the 1 K pot where monitored with silicon
diode thermometers and both the 3He pot and the cell had ruthenium-oxide
ones. The latter thermometers where calibrated at NIST and all tempera-
tures quoted in the thesis where converted from Ohms to Kelvin using this
calibration.
Finally, I built heaters using a design similar to that seen for the thermome-
ter in figure 4.10A. Specifically, I wound a length of highly resistive 0.1 mm
manganin wire tightly around a post, such that driving a current through the
84 Sample Preparation and Flow Measurement Technical Details
wire would warm it up via Joule heating, P = Rheater ·I2. The voltage used to
produce heat in these bobbins can be precisely controlled with the PID loop,
as long as a thermometer is positioned nearby.
85
Chapter 5
Flow Conductance of a Single
Nanopore
5.1 Mass Flow in a Single Nanopore
The first mass transport experiments I present in this chapter were performed
to test our experimental scheme and determine whether a quantitative mea-
surement could be achieved with a mass spectrometer. This proof-of-concept
[91] set the foundation for the analysis of all subsequent mass transport mea-
surements. The first sample for which the mass flow was investigated (shown
in figure 4.2B) had a diameter of 101±2 nm as measured from the TEM image,
and a membrane thickness of 50 nm. These measurements were conducted in
a first-generation experimental cell similar to the one shown in figure 4.5. As
explained in detail in chapter 4, the conductance G of a channel can be deter-
mined from the mass flow Q measured as a function of the pressure applied
across both sides of that channel, ∆P = P top − P bottom. Keeping the bottom
86 Flow Conductance of a Single Nanopore
reservoir of the experimental cell at zero pressure, P bottom ≈ 0 ensures that the
pressure differential is in practice equal to the pressure of the inlet (source)
reservoir above the membrane, ∆P ≡ P top−P bottom ≈ P top. Varying P top then
induces a change in the mass flow, i.e. P top1 → P top
2 ⇒ Q1 → Q2 that we can
detect with the Smart Test leak detector.
5.1.1 Measurement of Helium Gas Flow
Using the measurement scheme described in the previous chapter, I measured
the mass flow through the 101 nm nanopore for a series of pressure differential
applied across the sample at temperatures between liquid nitrogen and room
temperature. When changing the pressure in discrete steps from one set-point
to another, the signal measured changes asymptotically towards the next stable
value. In other words, the mass flow signal has a transient time of equilibration
before the system stabilizes, leading to an inverse exponential (equation 5.1)
behavior after the pressure is increased in a single finite step.
Q(t, P ) = Qi(Pi) + Ai · e−t/τ (5.1)
The equilibrium time constant τ varies between a few seconds to almost 2000
s depending on the experimental configuration used. For example, the equili-
bration time constant of the results in figure 5.1 was 17.5±0.8 s. The addition
of cold filters and longer capillaries in recent modifications to our apparatus
increased the impedance to the flow of gas going towards the leak detector.
This resulted in much longer equilibration times for each flow measurements,
and thus reduced the total number of data points that could be obtained.
5.1 Mass Flow in a Single Nanopore 87
Typically, after a duration equivalent to ∼2τ , the mass flow Qi corresponding
to the applied pressure Pi can be determined by applying a non-linear fit to
the time-dependent signal. An example of this is shown in figure 5.1, where
the red line is a portion of the exponential decay curve fitted to the mass flow
data (dots). In this example, the data show stepwise decreases in pressure
leading to descending stair-like mass flow signal as a function of time. This
procedure can be repeated multiple times, either in ascending or descending
pressure steps.
Fig. 5.1 Volumetric flow through a single 101 nm nanopore dur-ing an experiment as the pressure differential is decreased in step-wise fashion. When the pressure is decreased and left at a stablevalue, the response of the flow signal exponentially approaches anew equilibrium value. In red is the non-linear fit of equation 5.1used to extract the equilibrium value at a given pressure.
Once all data points Q(Pi) are extracted from the time-dependent signal,
88 Flow Conductance of a Single Nanopore
the mass flow as a function of applied pressure is explicitly obtained (see figure
5.2). These measurements are, however, subject to a systematic uncertainty
caused by a residual “background signal” in the mass spectrometer. This signal
offset is corrected by subtracting a constant to all flow signal and, in effect, en-
forces the condition Q(P = 0) ≡ 0. In general, the absolute value of the offset
depends on the state of the leak detector, but is normally <5·10−10 mbar·l/s
(using the same units as in figures 5.2 and 5.3A). This systematic error is al-
ways corrected, even if it is typically, at most, a few percent of the smallest
flow measured. Indeed, the majority of flow measurements are several orders
of magnitude larger than this correction. The final error in the conductance
is usually dominated by the propagated uncertainty from the leak detector
signal, the error in the pressure, and the temperature reading.
One clear observation that can be made regarding the data in figure 5.2
is that they deviate from the linear relationship shown by the red line. This
indicates that the nanopore conductance changes throughout the experiment
as the internal pressure of the gas above the nanopore increases.
5.1.2 Conductance of a Single 101 nm Nanopore
Using the definition of conductance G ≡ Q/∆P , we can plot the nanopore
conductance as a function of the pressure applied, or equivalently as a func-
tion of the Knudsen number Kn= λ/D. Figure 5.3 shows the conductance of
the nanopore at 77 K for pressures up to 38 bar. The variation in mean-free-
path λ of nearly four orders of magnitude induces a clear change in the flow
regime through the sample. The three smaller graphs of figure 5.3A contain
5.1 Mass Flow in a Single Nanopore 89
Fig. 5.2 Mass flow as a function of the pressure applied on a101 nm diameter pore. The red straight line is a guide to the eyeto accentuate the non-linearity of the data.
the same mass flow data as in figure 5.2, but segmented to better illustrate
the flow behavior. The first distinct pressure range P ≈ 0 − 55 mbar (green
background) shows very good linearity whereas the range P ≈ 0.69 − 5 bar
(beige background) and P > 6.2 bar (blue background) show strong and weak
departure from linearity, respectively. Easy comparison to the conductance
data of figure 5.3B can be done using the background color corresponding to
each flow regime. The first observation one can make regarding the conduc-
tance data in figure 5.3B is that it is indeed non-constant as the properties of
the gas change. A constant conductance leads to a linear relationship between
pressure and flow as can be seen, for example, in the low pressure points on
the bottom left graph of Fig. 5.3A. This linearity is then broken for flows at
90 Flow Conductance of a Single Nanopore
larger pressure differentials. The detailed dependence of flow on the pressure
is easier to visualize if we further analyze the conductance of the pore.
Fig. 5.3 (A) Volumetric mass flow of helium gas for three differ-ent ranges of pressure. The color of the background correspondsbetween figure A and B to distinguish between flow regimes. Ingreen is the Knudsen effusion region, in blue is the continuum hy-drodynamics viscous flow region and in between (beige) is the tran-sition region. (B) Conductance as a function of Knudsen numberapplied on a 101 nm diameter pore. The dotted red line is theKnudsen effusion contribution, the green dash-dotted line is theviscous short-pipe laminar flow contribution and the dashed blueline is the total flow fitted to the data.
5.1 Mass Flow in a Single Nanopore 91
Extracting an Effective Radius
The constant conductance observed at high Knudsen number is consistent
with the Knudsen effusion for finite-length channels. As noted in chapter 3,
the short pipe conductance in the Knudsen regime is given by equation 3.6
and fitting it to the data above Kn = 10 with L = 50 nm gives a radius
of 50.0±0.2 nm. This is within 1% of the radius extracted from the TEM
image of the nanopore and is within its uncertainty. The high precision of
the radius one can extract by modeling the conductance of a single nanopore
in the free-molecular regime thus gives an in situ tool to confirm the status
of nanopores during experiments. Indeed, as I will show in subsequent sec-
tions, this same fitting procedure can be used to quantitatively determine the
nanopore dimension and could alleviate the need for TEM imaging.
Modeling of the Transition Flow
Given the accuracy of the Knudsen effusion fit to determine the radius of the
nanopore, we attempted to model the conductance over the whole transitional
region. In order to do this, we used a method similar to the one presented
in references [92] and [93]. The authors of these works developed a model for
channel flows when the Knudsen number is in the transitional region. When
this is the case, the continuum assumptions begin to break down close to the
walls, in the so-called Knudsen layer. Within this layer, interactions at the
walls are affected by rarefaction effects and the “no-slip” assumption on the
walls can no longer apply. The emergence of a Knudsen layer as the fluid
approaches Kn ≈ 10−2 has an effect equivalent to allowing a finite slip at the
92 Flow Conductance of a Single Nanopore
wall-fluid interface and a correction must be applied to the continuum equa-
tion. This correction is expected to scale with the Knudsen number throughout
the transitional region until the flow is completely dominated by the rarefac-
tion effects.
The authors in [92] have derived the channel flow rate expected in pipes
and ducts,
Q =πR4P
8η0kBT
∆P
L(1 + aKn)
(1 +
4
1− bKn
), (5.2)
where P is the average pressure in the channel and η0 is the dynamic viscosity
of the fluid in the bulk. The first term outside the parenthesis is essentially
the continuum pipe flow found in equation 3.7. The second parenthesis is a
correction to the diffusion (i.e. acting on the viscosity η) for the generalized
slip model. In theory, the breakdown of the continuum hypothesis, i.e. the
rise of a slip boundary condition, acts as a reduction of the effective viscosity
µ and an increase in the flow rate predicted with respect to the non-corrected
viscous flow model. The parameter b has been found to best fit molecular
dynamics simulations for a value of b = −1. The remaining term, within the
first parenthesis, is a rarefaction correction Cr = (1 + aKn), where a must
transition “from zero in the slip flow regime to its asymptotic constant value
in the free molecular flow regime”[92]. Equation 5.2 can thus be re-written in
terms of a viscous flow rate in the slip regime Qslip,
Qtotal = Qslip · Cr(Kn). (5.3)
5.1 Mass Flow in a Single Nanopore 93
Beskok et al. [92] have chosen a definition of the correction Cr such that Qtotal
remains valid throughout the transitional region. Specifically, the parameter
a acts as a smoothly increasing function of Kn of the form
a = a02
πatan(a1Knβ),
such that the total flow remains equal to Qslip at Kn 1 and reaches the
molecular free flow asymptotically with Kn increasing. The two free parame-
ters, a1 and β are fit parameters defining the curvature of the transition be-
tween the two regimes and a0 can be chosen as the value of the free-molecular
flow rate.
While we can use the same arguments to model the flow in our nanopores,
as explained above, the aspect ratio of our nanopores is too close to unity to
warrant the use of the long pipe approximation (Eq. 3.7). A better model in
the continuum regime, for short-pipes, is given by equation 3.10. Equivalently,
at the other limit of large Kn, we know the total flow must be equal to the
Knudsen effusion of equation 3.6. The smooth cross-over between the two
limiting flows can be accomplished with a correction similar to Cr. We define
Gtotal = Gshort−pipecontinuum +
2
πatan(σKn) ·Gshort−pipe
Knudsen , (5.4)
as the nanopore conductance over the whole transitional region (0.01<Kn<10).
Here, the normalization of the arc-tangent ensures an asymptotic conductance
of the nanopore at large Kn and the fit parameter σ smoothly modulates
the emergence of rarefaction effects in the nanopores. The first term on the
94 Flow Conductance of a Single Nanopore
right-hand side is the conductance of a pore described by equation 3.10. The
term Gshort−pipecontinuum also contains the parameter α, which can thus be treated as
a phenomenological constant specific to the solid-state nanoholes used in our
experiments.
Using equation 5.4 to fit the whole dataset shown in figure 5.3, we found
parameters α = 4.69 ± 0.06 and σ = 5.3 ± 0.1. In this case, α is larger
than what was predicted in macroscopic apertures [56], which may indicate
that more important corrections from the end effects are required. The second
parameter (σ) defines a smooth cut-off point where rarefaction effects dissipate.
A larger value in σ translates into a larger contribution from effusion in the
transition region. The dotted red line in figure 5.3 is the Knudsen component
(Gshort−pipeKnudsen ) of the total conductance. For completeness, the green dashed line
shows the viscous hydrodynamics-type component Gshort−pipecontinuum .
Our data are limited by the maximum pressure that can be applied before
inelastic deformations of the membrane occur. With this sample, the lowest
Knudsen number we could attain was Kn = 0.013. Strictly speaking, this is not
yet within the purely viscous laminar flow region and this limits the accuracy of
α. In principle, to reach the laminar viscous flow regime with no-slip boundary
condition, we need Kn ≤ 0.001, which means a pressure approximately 13
times higher than that which was used in the present experiment. This is
currently not possible with the solid-state membranes, it however represents
an alternative avenue for investigation if sufficiently sturdy membranes were
fabricated. Strickly speaking, the other option to reach Kn ≤ 0.001 is to
increase the diameter of the pores, but this is contrary to our stated goal of
5.1 Mass Flow in a Single Nanopore 95
investigating nanofluidics.
5.1.3 Temperature Dependence of Conductance
In order to deepen our understanding of the crossover between the hydrody-
namics of viscous flow and the free-molecular Knudsen effusion, we conducted
a thorough investigation of the temperature dependence of the gas flow be-
tween 77 K and 295 K. To achieve this, we slowly varied the temperature of
the experimental cell by exposing it to cold nitrogen gas above a liquid nitrogen
bath. The decrease in temperature of the gas within the cell induced a drop
in pressure that could be monitored with a 0.001psi precision. Since the gas
handling panel supplying the gas remains at room temperature, one can view
this system as two volumes VGHS and Vcell that are at different temperatures
but connected by a negligibly small capillary allowing particles through such
that at equilibrium, pressure is the same in each volume. This is in fact the
very basis of a gas thermometer [94, 95].
Fixing the initial pressure applied on the nanopore, the flow was measured
continuously as the cell was cooled down and warmed back up. The conduc-
tance was calculated for the whole temperature range such that the tempera-
ture dependence could be characterized. Two examples of these datasets are
shown in the inset to figure 5.4. Seen with log-log axes, one can visualize the
linear relationship characteristic of a power-law dependence on temperature
G ∝ T−n. A single exponent “n” was extracted from each similar dataset.
As noted above, the equilibrated pressure of the whole system decreases
monotonically with the temperature of the cell. Taking the specific example of
96 Flow Conductance of a Single Nanopore
Fig. 5.4 The power law exponent of the temperature dependenceof conductance in a single 101 nm nanopore. The mean Knudsennumber Kn is used to position the points on the Knudsen scale.The inset shows a log-log plot of the conductance for temperaturesbetween the liquid nitrogen boiling point and room temperature.The slope of these points gives the power law exponent shown inthe main figure. A dashed line is shown at 0.5 as a guide to the eyefor the 1/
√T relationship expected for Knudsen effusion.
the data with green diamonds (Fig. 5.4 inset), the initial pressure, in the case
where the whole system was at room temperature, was 6.60 bar decreasing
to 6.08 bar after cooling the cell to 77 K. These two pressures each have an
associated Knudsen number, Kn = 0.286 at T = 295 K and Kn = 0.083 at
T = 77 K. We chose to take the mean value Kn to plot the exponent n as
a function of Knudsen number. In effect, this means the dataset with green
5.1 Mass Flow in a Single Nanopore 97
diamonds in the inset of figure 5.4 gives a point at Kn = 0.18 and the one
with blue circles has Kn = 12.2. A series of similar curves was measured
for initial conditions spanning the whole transition region. The exponent in
the power-law dependence on temperature of conductance for all available gas
pressures are shown in figure 5.4. In the Knudsen effusion regime, a G ∝ 1/√T
is explicitly present and the corresponding exponent n = 0.5 is shown with a
dashed line on the graph.
The exponent extracted from the dependence on temperature shows clear
departure from a purely inverse square root behavior at Kn . 10. The ex-
ponent increases slowly until it peaks to n ' 0.67 at Kn ' 0.2 and then
starts falling off to n ∼ 0.58 for the highest pressure attainable in the exper-
iment. The fall-off behavior observed below Kn ' 0.2 can be interpreted as
the crossover towards a conductance dominated by finite-sized viscous flow.
Indeed, taking the large pressure limit of the conductance in the short-pipe
viscous flow (equation 3.10) and expanding the density of helium gas into its
dependence on pressure and temperature, we obtain a short-pipe conductance
limit at high pressure of G∆P→∞viscous = πR2
√m
αkBT. This asymptotic limit of the
conductance in short pipe has the inverse square root dependence on temper-
ature that figure 5.4 hints at. The lack of a pressure dependence in the latter
equation is expected to lead to a plateau for the conductance, unfortunately
our data did not quite reach this regime, as can be seen in the low Knudsen
number data points of figure 5.3.
98 Flow Conductance of a Single Nanopore
5.2 Knudsen Effusion in Smaller Pores
The positive results obtained with the ∼100 nm pore constitute a foundation
for measurements in smaller pores. We have subsequently measured the mass
flow in dozens of nanopores with progressively smaller diameters and selected
a few to demonstrate the applicability of our method as we move towards the
1D limit. The conductance of each of these samples was obtained and then
fitted with the model of section 5.1.2. Several of these results are presented in
the remainder of chapter 5. The selected fit and conductances are shown in a
way such that each graph illustrates a different aspect of the model, as well
as demonstrates the repeatability of our measurements. Whenever possible,
measurements were performed at both room and liquid nitrogen temperature,
and a fit with equation 5.4 was computed when the diffusion through the
epoxy was negligible. Whether measurements were achieved with increasing
or decreasing steps in pressure across the nanopores did not yield different
results and the direction of the pressure change is therefore not distinguished
in the following sections.
5.2.1 Applicability of the Method in Smaller Nanopores
Figure 5.5 shows the conductance of two nanopore samples with diameters of
77± 1 nm and 46.5± 1 nm, as measured from TEM images. Each sample has
its associated picture inset in the graph of the conductance. Both wafers used
to produce these nanopores had a thickness of 50 nm.
For the sample shown in A, the mass flow was measured at pressures be-
tween 4.7 mbar and 17.7 bar. The conductance inferred from that flow mea-
5.2 Knudsen Effusion in Smaller Pores 99
Fig. 5.5 (A) Normalized conductance of a 77 nm diameternanopore for 297 and 77 K. The conductance data are normalizedwith the experimental conductance in the Knudsen regime in eachdata set. The dashed line is computed with equation 5.4 fitted tothe 77 K conductance data. (B) Conductance at room temperaturefor the sample seen in the inset. The diameter from TEM is mea-sured to be 46.5± 1 nm. Overlaid on the data are the phenomeno-logical model equations with radii equal to the TEM dimension(blue) and ±1 nm for green and magenta curves, respectively. Thewhite bar on the TEM image is 50 nm for A and 10 nm for B.
surement was then normalized with the experimental conductance in the Knud-
sen effusion regime GKnexp. =3.78·10−18 m·s (1.91·10−18 m·s) for 77 K (295 K).
The two curves are qualitatively very similar and this demonstrates that our
modeling of the flow is not dependent on the temperature at which the gas
flow measurements were performed.
The fit to the conductance in 5.5A is shown with a dashed line. Here, the
blue circles are data at low temperature (77 K) and red diamonds at ambient
temperature (295 K). The Knudsen effusion predicted by equation 3.6 for a
cylindrical channel with a diameter of 77 nm is lower than the experimental
100 Flow Conductance of a Single Nanopore
conductance (for Kn > 10). We attribute this discrepancy to the fact that our
samples are not perfect cylinders and, as explained in section 3.2.1, the opening
angle of nanopores affects the Clausing factor, thus changing the conductance.
Using the opening angle as a fit parameter, we find an 11 angle for the sample
in figure 5.5A. Once this correction to the Knudsen effusion model is applied,
the conductance over the whole transition region can be fit with equation 5.4.
The parameters α = 3.1± 0.1 and σ = 13± 2 were found for the two datasets
of figure 5.5A. For technical reasons,related to the protection of the sample,
the dataset at low temperature spans a much broader region of the Knudsen
scale and only the fit to this data is shown.
The same procedure was applied on the sample shown in figure 5.5B where
only the room temperature data are available (because of the type of sealant
that was used to prepare the SiN wafer). The fit to the high Knudsen number
conductance had to be corrected again for the pore geometry and an angle
of 11.5 was required to fit the experimental flow. The green and magenta
curves in figure 5.5B are produced using equation 5.4 with a radius modified
by±1 nm, which corresponds to the uncertainty from the dimensions measured
on the TEM image. This comparison of slightly different radii demonstrates
the sensitivity of our measurement scheme. Specifically, a reduction in the
diameter of a nanopore by either contaminants or structural deformation of
the SiN membrane can have easily detectable changes in the conductance.
5.2 Knudsen Effusion in Smaller Pores 101
5.2.2 Evidence of Nanopore Deformation
In order to further investigate the sensitivity of the mass flow on the dimensions
of the pore, we have repeatedly measured the flow through samples of various
length-to-radius (L/R) ratio and over extended periods of time. Specifically,
we have extracted the conductance of a given sample for a series of time in-
tervals following initial drilling, and correlated these measurements with TEM
imaging. A summary of the results of this series of experiments is available in
appendix A and a selection of two samples is shown here to demonstrate the
main findings.
Figure 5.6A shows the conductance of a 41± 2 nm nanopore, while figure
5.6B shows a 25±1 nm diameter pore, as measured from TEM imaging directly
after fabrication. The thicknesses of these samples are 50 nm and 75 nm,
corresponding to L/R ratios of 2.4 and 3.0, respectively. As has been observed
elsewhere [96], the larger this ratio, the more likely it is that a deformation
of the nanopore will occur over extended periods of time. For the sample
shown in the inset of 5.6A, the conductance was measured at 77 K for two
consecutive days. The measurement shows a slight offset between the first day
(teal circles) and the second day (blue diamonds). This illustrates that the
typical amplitude shift of the conductance can be observed in the first few
days after fabrication for nanopores with comparable geometries.
The fitting of data at high Knudsen number yields a radius of 20.43 nm and
20.36 nm for the teal and blue data, respectively. This radius is within the un-
certainty from the TEM imaging and the small shift is within our expectations
for a local deformation of the nanopore walls. Monitoring the Knudsen effusion
102 Flow Conductance of a Single Nanopore
Fig. 5.6 Conductance of 41 nm (A) and 25 nm (B) diameternanopores as a function of the Knudsen number. Dotted lines areKnudsen effusion contribution to total conductance in A and a con-stant function fit the data at high Knudsen number. In both cases,a slight shift in the conductance is detected between measurementsmade several hours apart. The white bar on the TEM image is20 nm long in A and 10 nm in B.
in nanopores over time can thus be used to determine when the pore dimen-
sions become stable. When this is the case, subsequent experiments can be
performed reliably. This is important, since I have once observed a nanopore
gradually being blocked by contaminants and become fully obstructed during
a continuous measurement of mass flow in the Knudsen regime.
A more dramatic change was observed in the sample shown in the inset
of figure 5.6B (also shown in figure 4.2E) and whose room temperature con-
ductance in the Knudsen regime was found to drop by ∼21% over 48 hours.
These conductance data are shown in figure 5.6B, where the dotted lines are
fit for Kn >10 of 1.15 · 10−19 m·s and 0.94 · 10−19 m·s. This corresponds to a
decrease in fitted radius from 12.7 to 11.7 nm with a constant opening angle
5.2 Knudsen Effusion in Smaller Pores 103
of 5. The observation of such a sharp decrease in conductance for samples
with similar L/R led us to investigate the correlation between the nanopore
dimensions at the initial drilling and the structural stability of nanopores over
time (see appendix A). The most relevant conclusion to the current line of
experiments is that, in order to obtain stable nanopores with much smaller
diameters, we needed to use thinner membranes to keep the L/R ratio as low
as possible.
5.2.3 Conductance of the Smallest Single Nanopores
Once these new constraints in our fabrication protocol were better defined, we
were able to achieve gas flow measurements in much smaller nanopores. To
our knowledge, these are the smallest single nanopores characterized to date
using direct gas transport measurement. Figure 5.7 shows the conductance of
the sample shown earlier in figure 4.2F with a diameter ∼21 nm. An explicit
demonstration of the stability of this pore is visible in the inset, and these
measurements were taken at three and twelve days interval. The data was
taken at 77 K using the same procedure as was described earlier. The first
two datasets (red and blue dots of figure 5.7B) were obtained quickly as initial
verification of the pore dimensions. These datasets were also limited to high
Knudsen numbers in order to restrain the duration of the experimental runs.
The last series of points (black dots) were taken more carefully and for applied
pressures up to 56.8 bar. The data from this last set were measured over two
consecutive days and a slight vertical shift is observed around Kn = 0.12; this
is between the left-most points from the second day and the first day’s data
104 Flow Conductance of a Single Nanopore
extending to high Knudsen numbers.
Fig. 5.7 Conductance, at 77 K, of a nanopore with a 21 nmdiameter. The inset shows the TEM image of the pore. (B) Similarmeasurements were taken 12 days (red) and 9 days (blue) priorthe conductance shown with black dots. The black dots are thesame data for A and B. The stability over time is evident from therepeatability of the measurement. The white bar is 10 nm long andthe TEM image is also shown in 4.2F.
The middle curve (green) in figure 5.7A is the total conductance function fit
to the data and it is also reproduced with a radius 5 A larger (teal curve) and
smaller (crimson curve). Comparing to the effect predicted from a shift of a few
Angstrom, the slight offset observed during the second day of measurements
is not considered to be caused by a significant deformation of the nanopore.
Figure 5.7B shows the measurements are repeatable.
The parameters of the fits shown in figure 5.7A are σ = 18 and α = 9.0 for
a radius of R = 10.6 nm and an opening angle of 7. Comparing samples of
different sizes, we find a general trend that these parameters increase as the
nanopore samples decrease in size. However, a direct physical interpretation
cannot be easily attributed to this observation because of the limiting phe-
5.2 Knudsen Effusion in Smaller Pores 105
nomenological nature of the model. Figures 5.7 and 5.8 both show deviations
from the conductance equation over the transition region, however more sam-
ples need to be measured, over the full range of available pressures, before the
applicability of the short-pipe flow model can be assessed in greater detail.
The smallest sample I was able to measure is shown in the inset of figure
5.8. The TEM image shows that the nanopore is slightly elongated and the
length measured along the longest axis is 15.6 nm and 14.2 nm along the
shortest. The wafer for this sample was 30 nm thick.
Fig. 5.8 Conductance of a nanopore 15 nm in diameter. The solidblue line is the total conductance equation fit to the data and thedotted lines on either side are computed from the same equationwith angles ±1. The inset shows the TEM image with a 10 nmlong white bar for calibration.
The mass flow through this sample was measured from 9.0 mbar to 7.2 bar
and in a liquid nitrogen bath at 77 K. The solid line overlaid on the data points
is the total flow equation with α = 8.5 and σ = 7.0 for radius of R = 7.8 nm
106 Flow Conductance of a Single Nanopore
with an angle of 2.7. The two dotted lines show the effect of varying the
opening angle by 1. These results confirm that a short-pipe Knudsen effusion
model can quantitatively predict the conductance in single nanopores as small
as ∼15 nm in diameter.
However, the feature seen around Kn = 1.5 could not be explained by a
specific experimental event and its origin remains unexplained. I attempted
to repeat the mass flow measurements using the same experimental conditions
and the plateau feature was observed again at the same pressure, returning to
the smooth curve towards higher Knudsen numbers. Unfortunately, this is not
sufficient evidence to attribute a physical origin to this irregularity, since such
a deviation was not observed in the course of measurements of other samples.
5.3 Conclusions from Gas Flow Measurements
We have measured the mass transport of helium gas through single nanopores
fabricated in free-standing SiN membranes. The conductance of the nanopores
was calculated from the observed flow for a wide range of pressures and tem-
peratures. A clear transition was observed as the system transitioned from
free-molecular effusion at low pressure to viscous laminar flow described by
the hydrodynamics of continuous media. This transition was characterized in
nanopores with diameters as small as 15 nm. The conductance in the Knud-
sen regime was found to be highly sensitive to changes in the dimensions of
the nanopores as well as structural deformations of its walls. Prolonged mon-
itoring of the conductance of a nanopore has been used to detect when the
nanopore becomes structurally stable. Using our mass spectrometry measure-
5.3 Conclusions from Gas Flow Measurements 107
ment scheme has also allowed us to successfully detect the mass transport in a
single 15 nm diameter nanopore, which is the smallest nanopore to date where
such a measurement has been achieved. Furthermore, the characterization of
the flow in single nanopores establishes a foundation for the quantum fluid
flow experiments presented in the next chapter.
108
109
Chapter 6
Hydrodynamics of Superfluid
Helium in a Single Nanohole
This chapter presents the results we obtained by extending our measurement
scheme to temperatures near the absolute zero. The measurement of mass
flow was achieved in an experimental cell mounted inside a cryostat, as de-
scribed in section 4.4. All mass flow measurements were performed with a
Pfeiffer Smart Test mass spectrometer and the flow rate was calibrated before
every experiment with an external calibrated leak with a constant leak rate of
4.5·10−3 ± 10% ng/s.
The first sample in which superfluid flow was observed is shown in figure
4.2D. It had a nearly circular cross-section with dimensions of 42.7 and 45.6 nm
along the shortest and longest axis, respectively. As with previous samples,
the wafer was etched to leave a square free-standing silicon nitride window
of ∼30 µm wide and has a thickness of 50 nm. This sample is very similar
110 Hydrodynamics of Superfluid Helium in a Single Nanohole
to the one presented in figure 5.5B, albeit with a slightly smaller diameter.
Using the Knudsen effusion in the gas phase, we verified that the sample
dimensions remained the same throughout the experiments with superfluid.
This first sample was measured in an experimental cell of the first generation
(see section 4.3.1) and our results with this sample were published in Physical
Review Letters [97].
The clean helium gas coming from the gas handling system was delivered
into the cryostat where it passed through several thermal anchors with decre-
mental temperature stages. Under normal operation, the temperature of these
stages was set below the liquefaction point such that the liquid-gas boundary
remains outside the experimental cell. The source side of the cell could there-
fore be kept full of liquid helium during the experiments. The sintered silver
powder inside the thermal anchors offered the large surface area required for
good thermal contact between the superfluid and the metallic surfaces of the
enclosures. Details on the thermometry and temperature control have been
given in chapter 4. The equilibrium time constant for the stabilization of each
flow measurement in the cryostat increased to approximately 600 to 1200 sec-
onds. This limited the number of points that could be obtained during each
experimental run, but was sufficiently short for a careful measurement to be
achieved.
6.1 Knudsen Effusion Below 20 Kelvin
The first gas flow measurement made on this sample were completed at room
temperature on two separate days for low and high pressure ranges respec-
6.1 Knudsen Effusion Below 20 Kelvin 111
tively. The first day of measurement was performed over the range 30 to
790 mbar pressure, and the second day covered pressures up to 22.8 bar. Fig-
ure 6.1A shows the conductance of the sample at 295 K with the usual fit
of the phenomenological model for the conductance in the transition region.
Sister curves were also added for radii of ±1 nm. The parameters of this fit
are α = 4.7 and σ = 11 for a radius of 22.1 nm with an opening angle of 9.
The inset shows the TEM image taken during fabrication.
1000
Con
duct
ance
G [
10-1
8 m s
]
Knudsen number Kn
±0.5 nmR=21.4 nm
1001010.10.45
0.50
0.55
0.60
0.65
0.70
0.75
.
1
100101
2
3
0
R=22.4 nm10 K
±0.5 nm
20 K
Knudsen number Kn
Con
duct
ance
G [
10-1
8 m s
].
A B
Fig. 6.1 (A) Conductance at room temperature of a singlenanopore with a 45 nm diameter. The solid line is a fit of thetotal conductance equation with radius R = 22.1 nm. The twodotted lines represent the same function, but with radius largerand smaller by 1 nanometer. The white bar is 50 nm. (B) Similarconductance data taken at 20 K (teal dots) and 10 K (magentadiamonds). The lines over these datasets are from the Knudseneffusion conductance with R = 21.4 nm and R = 22.4 nm for 20 Kand 10 K. In this case, the dotted lines are for a radius ± 0.5 nm.
Figure 6.1B shows conductance measurements at much lower temperatures,
specifically at 10 K and 20 K. This data was obtained after all of the super-
fluid experiments were performed and demonstrates that the nanopore re-
112 Hydrodynamics of Superfluid Helium in a Single Nanohole
mained structurally stable throughout the several months of operation. The
fit of Knudsen effusion conductance to the data at low temperature give radii
of R = 22.4±0.3 nm and R = 21.4±0.4 nm for the 10 K and 20 K data re-
spectively. Additional dotted lines are shown for expected conductance using
radii ±0.5 nm.
6.2 Liquid Helium Mass Transport
The typical procedure to determine the superfluid mass flow is as follows.
First, both sides of the experimental cell are emptied at a temperature higher
than the helium boiling point. Then, the apparatus is cooled below the super-
fluid transition such that the introduction of helium gas, via the gas handling
system, forces condensation within the heat exchangers. This liquefaction is
a slow process and only once all of the volume is filled with liquid helium
does the data collection really begin. The helium atoms that pass through the
nanopore are pumped out of the cryostat by the mass spectrometer, where they
are detected. Keeping the applied pressure constant, the mass flow through
the nanopore was monitored as the temperature was increased incrementally
from 1.6 K to 2.5 K or more. Several such data sweeps are shown in figure 6.2.
6.2.1 Viscous Fluid Flow
In figure 6.2, the dotted line labeled Tλ is the bulk superfluid transition tem-
perature. On the right-hand side of this line, there is no superfluidity expected
to occur and only “normal” liquid helium should be present in the cell. In this
portion of the graph, the data from each set show a clear increase in mass flow
6.2 Liquid Helium Mass Transport 113
1.6 2.0 2.4 2.8 3.20
1
2
3
4
5
6
Tλ
Mas
s flo
w [
ng/s
]
1.45 bar0.4830.3450.2410.1450.069
Temperature [K]
Fig. 6.2 Mass flow of liquid helium through a single 45 nmnanopore. Each symbol corresponds to a distinct temperaturesweep at a fixed pressure of 0.069, 0.145, 0.241, 0.345 and 1.45 bargoing from bottom to top. The dashed line shows the superfluidtransition temperature for bulk helium (Tλ).
in response to an applied pressure gradient. This pressure-dependent mass
transport demonstrates that the viscous fluid flow of liquid helium is not neg-
ligible, and it indicates the normal component of helium, below the superfluid
transition, should also be taken into account. This is counter-intuitive and
we initially had made the naive assumption that this would not be the case.
Well known experiments showed decades ago that the normal component can
be filtered-out in channels with dimensions of the order of 10−7 m, and the
signals from the superfluid component could be measured exclusively. The
conductance of the nanopore used in this experiment is however large enough
114 Hydrodynamics of Superfluid Helium in a Single Nanohole
that the normal component remains a significant portion of the total flow. We
believe the normal component was able to flow because of the relatively small
length of the pore (L = 50 nm, D = 45 nm).
The theoretical framework used to treat the viscous component of the mass
transport was developed in chapter 3. This model made use of the short-
pipe viscous equation (Eq. 3.10) to determine the liquid helium flow through
a nanopore. This equation was used to fit the data at temperatures above
Tλ. The average of the radii found from fitting each dataset of figure 6.2 was
R = 20±2 nm, and an example of this fit is shown in figure 6.3 for clarity. This
average radius is in excellent agreement with the gas flow measurements and
the TEM imaging. The viscosity and density of liquid helium were taken from
the literature [61, 98, 99], and the length of the pore taken as the membrane
thickness of 50 nm. The α parameter for the end effects in the nanopore was
set to α = 4.7 as found in our earlier experiments with gas flows (see figure 6.1).
The modeling of the viscous flow was then used to determine the superfluid
component, as will be described below.
6.2.2 Superfluid Flow Through the 45 nm Nanopore
The data of figure 6.2 shows a clear increase of the mass flow when the tem-
perature decreases below the superfluid transition, and this is observed for all
pressures investigated. To better illustrate this increase, we focus our atten-
tion on a single temperature sweep shown in figure 6.2. Taking the curve for
483 mbar and reproducing it in figure 6.3, we easily notice the different flow
regimes on either side of Tλ. Here, the temperature axis was offset for clar-
6.2 Liquid Helium Mass Transport 115
ity in an attempt to alleviate the small shift in the superflow onset observed
in some of the data sets. We suspect that the temperature recorded by the
thermometer during the lower pressure experimental runs could have suffered
from a small gradient in the temperature across the experimental cell, which in
turn could have offsetted the temperature scale. The largest offset we recorded
was 84 mK for the 241 mbar series and, for the other datasets, an average of
32 mK shift. This correction to absolute temperature was applied before any
subsequent modeling and determination of the superfluid component.
-0.5 0.0 0.50
1
2
3
Mass
flo
w [
ng/s
]
[K]T−T*λ
Fig. 6.3 Mass flow measured at a constant pressure of 483 mbar.The connecting line is a guide to the eye. The green dotted lineis the flow computed by the short-pipe equation of viscous flow,using the total helium density ρ over the whole temperature range.This assumption effectively treats both He I and He II as classicalfluids. The red solid line is built from the same equation, but usingthe density of the normal component only. The temperature axis isdefined with respect to the superfluid transition temperature (Tλ).
116 Hydrodynamics of Superfluid Helium in a Single Nanohole
Total Flow
The equation for the two-fluid flux (equation 3.12) can be used to understand
the flow in the single nanopore. Assuming an average velocity in the pore,
we consider the total mass flow given by Qtotalm = JtotalπR
2 and the normal
component of He II described by a viscous short-pipe flow, as discussed before
(Eq. 3.10). The total mass flow is therefore given by
Qtotalm = πR2ρsvs +
8πηL
α
(√1 +
αρnR4
32η2L2∆P − 1
), (6.1)
and it can be compared to the mass flow measured experimentally. The only
unknown remaining in this equation is the superfluid velocity vs, which is
expected to be limited only by an upper-bound critical velocity.
In an effort to describe the observed mass flow through the 45 nm nanopore,
the right-hand side term of equation 6.1 was fit to each data series shown in
figure 6.2. Again focusing on the data in figure 6.3, we produced two curves
using the viscous short-pipe flow model (Eq. 3.10). The dotted green line
was computed using the total helium density for the whole temperature range,
thus working under a “no superfluid” assumption. Above Tλ this model agrees
well with the data, as expected for a purely “classical” viscous fluid. Below
Tλ, the observed flow is approximately twice as much as that which a simple
fluid would produce. We attribute this excess to a superfluid contribution
to the total flow. In contrast, the solid red line was computed using the
tabulated density from the normal component of the two-fluid model. This
normal density is similar to the values shown in figure 3.4 for bulk helium
at saturated vapor pressure. In effect, this red curve is the second term of
6.2 Liquid Helium Mass Transport 117
equation 6.1.
The superfluid flow was then extracted by subtracting the normal compo-
nent from the total flow. As mentioned in chapter 3, superfluid flow takes
place at a critical velocity beyond which it cannot cross because of the on-
set of energy dissipation. The critical velocity in other flow experiments is
usually reached at pressure heads of ∼O(1) Pa and so we therefore assumed
that the large pressure differential used in our experiments were sufficient for
the superfluid to reach the critical velocity. From the superfluid mass flow,
we can isolate the effective velocity of the superfluid at each point since the
cross sectional area and density are known. This velocity is interpreted as the
superfluid critical velocity and is discussed further in the following section.
6.2.3 Critical Velocity in a Nanopore
The critical velocity data inferred from the total mass flow is shown in figure
6.4. We note a clear dependence on the temperature and a nearly linear
regime for the points at lowest temperature. In figure 6.4, the critical velocities
were normalized with vc0, the extrapolated critical velocity at T = 0 K. This
extrapolation is accomplished using a linear fit of the critical velocity data.
The normalization makes it easier to compare to data from Zimmermann et
al. [100, 101]. In their work, they found that the dependence on temperature
of the critical velocity of superfluid in small channels follows a linear function
of the form vc(T ) = vc0(1−T/T0) (see also the discussion of equation 3.14). In
their experiments, vc0 was found to be of the order of 1 to 25 m/s [70, 100]. For
the single nanopore flows shown in figure 6.2, this extrapolation to T= 0 K
118 Hydrodynamics of Superfluid Helium in a Single Nanohole
gives values of vc0 between 8 and 45 m/s. It is worth mentioning that our
experiments are for continuous flow, whereas most results from other groups
are obtained with an oscillating membrane inducing an “AC” flow of very small
amplitude. This qualitative difference is especially important with respect to
the temporal resolution of the measurements; we observe long time averages
of the signal, whereas they can detect milliseconds dynamics of the flow.
[K]T-T*λ
0.0-0.8 -0.40.0
0.2
0.4
69241345
483 mbar
v c/vc0
Fig. 6.4 Critical velocity of the 45 nm nanopore as a functionof temperature. Each different symbol corresponds to the dataof figure 6.2 and the connecting lines are guides-to-the-eye. Eachdataset is normalized with vc0, the critical velocity extrapolated toT= 0 K (see text). This was done in order to compare to datafrom [100, 101] (open triangles). The error bars are defined bythe propagated uncertainty of the density, viscosity, flow and crosssectional area of the pore.
The error in the critical velocity is largest near Tλ because of the uncer-
6.3 Liquid Helium Mass Transport in 16 nm Nanopore 119
tainty of the superfluid density. This large error prevents us from resolving
the behavior of the critical velocity in the vicinity of the superfluid transition.
We however clearly observe that the temperature dependence of the superfluid
critical velocity shows similar behavior to what was found in larger channels
by other experimental techniques. To our knowledge, the absolute value of the
critical velocity inferred from our mass flow measurements is the largest ever
measured in channel flow experiments.
6.3 Liquid Helium Mass Transport in 16 nm Nanopore
Based on the results obtained in the 45 nm sample of the previous section,
we attempted to reproduce these measurements in smaller nanopores in order
to map out the behavior of liquid helium as it approaches the 1D limit. The
smallest sample that yielded a superfluid signal at low temperature is shown
in figure 6.5. The picture in the left panel (A) was taken after the sample
fabrication, which took place six months before the superfluid measurements
were conducted. The picture on the right panel of figure 6.5 (B) was taken
four months after the measurements, in order to demonstrate the long term
structural stability of the sample. The TEM images show a decrease of the
diameter from 17.3± 0.8 nm to 15.6±0.7 nm; this nearly identical dimension
is characteristic of a nanopore that has reached a fully stable configuration.
The membrane in which this sample was drilled was 30 nm thick and with the
usual free-standing surface of ∼30 µm by ∼30 µm.
The low temperature measurements with this sample were achieved in an
experimental cell of the second generation, and additional precautions were
120 Hydrodynamics of Superfluid Helium in a Single Nanohole
Fig. 6.5 (A)Image taken six months before the measurements ofsuperfluid flow in this nanopore and (B) the TEM image of the samesample taken four months after the measurements. The white baris 10 nm long.
taken to ensure ideal thermal anchoring and temperature monitoring. The
experimental procedures for the superfluid measurements of this sample were
replicated as closely as possible to what was described previously. The empty
cell was kept below liquefaction temperature and filled slowly with helium gas
until the condensation within all cold volumes was completed and an equilib-
rium was reached. At that point, the pressure applied above the membrane
can easily be kept constant. Setting this pressure at a desired value, the tem-
perature was then increased in a stepwise fashion in order to provide sufficient
time for the mass flow signal to equilibrate at each step. While the experimen-
tal protocol was similar, the modifications to the apparatus caused a drastic
increase in the typical time required for the signal to equilibrate. A single
measurement of the flow was now on the order of 6000 seconds. As was the
case with the larger 45 nm sample, a Knudsen effusion measurement was also
6.3 Liquid Helium Mass Transport in 16 nm Nanopore 121
performed in order to confirm the nanopore dimensions in situ.
6.3.1 Knudsen Effusion Measurement to Confirm Nanopore Radius
Gas effusion measurements were taken with the experimental cell at a con-
stant temperature of 20 K. Because of several changes to the apparatus imple-
mented prior to our measurements with this sample, it was necessary to test
whether the warm, low-density gas from the gas handling system was prop-
erly thermalizing in the newly installed thermal anchors. In effect, helium
at room temperature was cooled to ∼4 K while passing through the anchors,
then warmed back to 20 K immediately before entering the experimental cell.
Specifically, the thermal anchor on the 1 K pot were first kept at T < Tboiling
(3.8 K) while the gas pressure was varied to take the Knudsen effusion data.
In a second run, the thermal anchor was then kept at T > Tboiling (∼5 K), thus
preventing condensation of the helium gas. Two datasets of effusion at 20 K
were thus taken, at a two-days interval, to test whether the gas was properly
cooled before reaching the experimental cell.
The results of the two runs are shown in figure 6.6. These results are effec-
tively identical and, qualitatively, the flow exhibits no difference between the
two attempts. This indicates that the gas is indeed thermalized appropriately
by the last thermal anchor and the temperature of previous components of the
cryostat does not affect the measurement further down. The flow detected was
also quantitatively consistent with the Knudsen effusion theory for flow in a
single 16 nm nanopore, as is evident from the lines in figure 6.6. Because of
operational constraints in the laboratory, these gas flow measurements could
122 Hydrodynamics of Superfluid Helium in a Single Nanohole
not be achieved at 77 K, as was usually done with other samples. Nonetheless,
a useful comparison can be done with the effusion data of the other 16 nm
sample in the previous chapter (figure 5.8). By scaling-down the 20 K flow
with the square-root dependence on temperature of Knudsen effusion, we con-
firm the two samples yield comparable conductances. This simple comparison
quickly confirms our measurements are consistent between the two nanopores
with a radius ∼16 nm.
Con
duct
ance
G [1
0-1
8 m
s]
Knudsen number Kn
1001010.1
0.20
0.30
0.40 Day 2Day 1
fit (R=8.2 nm)±0.5 nm
.
Fig. 6.6 Gas conductance at 20 K of a 16 nm diameter nanopore.The two data series are for identical experiments realized two daysapart. The solid line is a fit to the data for the total conductancewith a radius of 8.2(1) nm, α = 4.7(1) and an opening angle of8. The dashed lines are from the same equation computed with aradius set at 7.7 nm and 8.7 nm.
The solid line overlaid on the conductance data at 20 K is the equation 5.4
with radius of R = 8.2± 0.5 nm and an 8 opening angle was included in the
determination of the Clausing factor. The two dashed lines on either side of
6.3 Liquid Helium Mass Transport in 16 nm Nanopore 123
the fit are computed using the same equation with radius larger or smaller by
0.5 nm. The confirmation of the radius from the conductance in the Knudsen
effusion regime gives us confidence the experiments in the superfluid phase can
be carried out, and that the sample dimensions can remain stable over time.
6.3.2 Superfluid Mass Flow in a 16 nm Nanopore
Measurements of liquid helium flow through the 16 nm sample are shown in
figure 6.7, where each set of colored symbols represents a constant-pressure
series of measurements. The solid lines between similar symbols are guides-
to-the-eye. Starting near T ' 1.55 K, the cell was warmed incrementally with
a given pressure head of 68.9, 145, 276, 345, 483 or 689 mbar. Sufficient time
was allowed for the mass flow signal to reach its asymptotic equilibrium, as
explained earlier in section 5.1.1. The signal from the mass spectrometer was
calibrated with the same external calibrated leak of 4.5 · 10−3 ng/s and the
sample was kept below T = 5 K during all liquid helium experiments.
In figure 6.7 the error bars show the uncertainty associated with the pre-
cision of the mass flow signal, and are typically smaller than the symbol size.
This is however not the largest source of uncertainty, as can be seen from the
689 mbar dataset. The few green triangles symbols not connected by the solid
line were taken on a separate day and show a slight offset that we attribute
to the uncertainty on the accuracy of the measurement set up. While this
systematic error remains small relative to the overall flow, it is larger than
other errors in several of the datasets. In order to keep the figure clear and be
able to observe the trend of mass flow across Tλ, the error bars used did not
124 Hydrodynamics of Superfluid Helium in a Single Nanohole
include this hard-to-estimate systematic uncertainty.
1.5 2.0 2.5
0.1
0.2
0.3
0.4
0.5
68.9145276345483689 mbar
Mas
s F
low
[ng
/s]
Temperature [K]
Tλ
Fig. 6.7 Superfluid mass flow through a 16 nm sample. Thedashed line is the superfluid transition temperature Tλ. The solidlines are guides-to-the-eyes.
A sharp change in flow regime is observed in the mass transport data at
the Tλ line (dashed line). The flow transitions from a nearly constant value to
a steadily increasing one when the temperature is lowered across Tλ. At the
lowest temperatures we reached, the flow is nearly three times larger than at
T = Tλ.
An important observation to make regarding these data series is that the
small discrepancy between the expected superfluid transition temperature and
the observed one is absent. This suggests that the small shifts observed with
the previous sample (section 6.2) were likely thermometry artifacts and the
6.3 Liquid Helium Mass Transport in 16 nm Nanopore 125
improvements to the apparatus, between the first and second generation of
experimental cells, provided the required thermal contact between the liquid
and the thermometry equipment.
The fitting procedure described earlier for the 45 nm sample has been
duplicated on these datasets. The viscous helium contribution to the total
flow was computed using the mass flow predicted from this equation, with
the density and viscosity interpolated from isobaric curves [61, 98, 99]. As an
example, the 483 mbar dataset was selected to show the result of the fit, as
seen in figure 6.8. The radius found from this fit was 8.3 nm, which is within
the 16± 1 nm diameter measured from TEM imaging. The α parameter used
was 4.7, which was obtained from the gas conductance measurements shown in
the previous section; this is also the value used for the larger 45 nm nanopore.
The fit of the short-pipe viscous flow model (Eq. 3.10) to the T > Tλ
data of the other datasets was also achieved. The radius found this way had a
larger variance than expected, compared to the similar fits done with the 45 nm
sample in the previous section. To be specific, for smaller pressures datasets,
the fitted radius was too large, and in the worst case, for the 69 mbar dataset,
the fit yielded a radius of 12 nm. This is well over the confidence interval
of our TEM diameter measurement. Given that the equation used in the
fit yielded worst results at low pressures, we examined the flow behavior as
P→ 0. In order to directly investigate this discrepancy, we measured the flow
as a function of the applied pressure for a fixed temperature of 2.25 K, well
within the He II region. The data we obtained is shown with black dots in
figure 6.9.
126 Hydrodynamics of Superfluid Helium in a Single Nanohole
1.4 1.6 1.8 2.0 2.2 2.4 2.60.0
0.1
0.2
0.3
0.4
Mas
s F
low
[ng
/s]
Temperature [K]
Fig. 6.8 Superfluid mass flow at 483 mbar in the 16 nm sam-ple. Shown as a dashed line is the contribution from the normalcomponent of liquid helium computed from the viscous short-pipeequation. The difference between the measured flow and the pre-dicted flow is assigned to the superfluid component.
The first observation one can make looking at these data is that the flow
appears to have a finite value as the pressure approaches zero. This is not
the behavior our model predicts and explains why the fit yielded much larger
radii for the lowest pressures. The red curve shows the viscous flow model of
equation 3.10 with a radius of 8.1 nm. The distribution of the data over this
curve is imbalanced, which indicates the equation might not describe the data
as accurately as possible. The simplest hypothesis one can make to explain this
discrepancy is to add a constant offset to the equation, which could account for
the “supplemental” flow observed. This offset would imply that a secondary,
parallel transport mechanism might be contributing to viscous flow. We fit
this modified model to the data, and the black curves in figure 6.9 show the
6.3 Liquid Helium Mass Transport in 16 nm Nanopore 127
0 500 1000 1500 20000.0
0.1
0.2
0.3
0.4
Flo
w [
ng/s
]
Pressure [mbar]
Eq. 3.10Eq. 3.10 + offset
Fig. 6.9 Liquid mass flow through a 16 nm sample as a functionof the applied pressure. The dots are the measured flow and theopen circles show the former values with a constant subtracted torespect Q(P=0)=0. The solid line is a fit to the data with a radiusof 7.7±0.4 nm.
result. The fitted radius was 7.7±0.4 nm, and the solid line shows the curve
with R = 7.7 nm, whereas the dashed lines on either side were computed with
R±0.4 nm. While it is interesting to note that a small offset can improve the
fit to the data, we do not have sufficient information to conclude on the validity
of this correction. This new radius fitted to the data remains consistent with
the dimensions measured by microscopy.
In summary, the viscous helium flow Qviscousshort−pipe can be modeled with the
short-pipe viscous equation. Here again, we attribute the difference between
the observed flow and the flow predicted by this equation to a superfluid con-
128 Hydrodynamics of Superfluid Helium in a Single Nanohole
tribution. As was done for the 45 nm nanopore, earlier in this chapter, the
effective superfluid velocity was inferred from this superfluid mass flow. The
superfluid contribution was extracted from the data in figure 6.7, and the ve-
locities inferred from them are shown in figure 6.10. The first panel (A) shows
the absolute velocities and the panel on the right (B) shows the results of the
483 mbar dataset normalized to the extrapolated T = 0 value vc0. This nor-
malization was performed to facilitate the comparison to data of [100] shown
as open triangles. In the determination of a critical velocity, we again make
the implicit assumption that the superfluid motion is limited by the emergence
of energy dissipation mechanisms.
Fig. 6.10 Critical velocity of the superfluid component in theliquid helium transport through the 16 nm pore. The symbols arethe same as in figure 6.7 and each corresponds to a given pressurehead. The vertical dotted line is positioned at T = 2.17 K to showthe superfluid transition temperature. (B) Critical velocity of thedata at 483 mbar and normalized to the extrapolated velocity atT = 0 K. Open triangles are data reproduced from Zimmermannet al.[100].
6.4 Critical Velocities of Superfluids in Single Nanopores 129
These results are qualitatively similar to those found for the 45 nm sam-
ple. Qualitatively, the superfluid contribution continues to exhibit a linear
dependence on temperature down to the lowest temperatures accessible with
the modified apparatus. Specifically, the critical velocity of the superfluid in
the 16 nm nanopore reaches this nearly linear regime below approximately
1.9 K. The behavior of vc is is easily compared to the data of [100], (see fig-
ure 6.10B), where both datasets demonstrates a behavior consistent with each
other. Quantitatively, the extrapolated critical velocity at zero temperature
were found to extend from 27±1 to 39.5±1.6 m/s for the 68.9 to 689 mbar
datasets respectively. Once again, in absolute values, these are larger than the
ones measured in micropores.
6.4 Critical Velocities of Superfluids in Single
Nanopores
The critical velocity inferred from the total flow of liquid helium in the two
samples shown in this chapter can easily be compared to other experiments
of superfluid flow. Figure 6.11 contains a compilation [64] of critical velocities
found over several decades and in a multitude of experiments (open symbols).
The red bars show the range of superfluid critical velocities found in our study
(values at T = 1.7 K for each data set). Our results are positioned in channel
sizes never before accessible and are the highest superfluid critical velocity ob-
served yet in channel transport experiments. In other words, our experimental
method has allowed us to determine the critical velocity of superfluid helium
in the smallest channels yet.
130 Hydrodynamics of Superfluid Helium in a Single Nanohole
Channel Diameter [m]
Crit
ical
Vel
ocity
v c [m
/s]
101
10-1
10-3
10-5
10-310-510-7
10-310-510-7
Fig. 6.11 Consolidated graph of critical velocities found by sev-eral groups over many years. The open circles are identified as in-trinsic critical velocities, whereas the other symbols have the chan-nel size dependent critical velocity closer in behavior to the Feyn-man model of quantized vortices (solid line). The red bars show thespan of critical velocities obtained with the two samples presentedin this chapter.
The open circle symbols are results of experiments where the superfluid
critical velocity was found to be size-independent and linearly dependent on
temperature. This class of critical velocities is quantitatively much larger than
the second type observed, diamonds (for older data) and triangles, those being
for critical velocities dependent on the dimensions of the flow channel. The two
types of critical velocities are referenced to as internal or external, in reference
to their dependence on the internal temperature or external pore geometry.
6.5 Summary 131
The physical phenomena driving the dissipation of energy is qualitatively dif-
ferent in both cases. For example, internal critical velocities have only been
observed in smaller pores where the temperature dependence suggested a ther-
mally activated process. This has been described as a stochastic nucleation of
phase slip events removing quantized circulation units [64]. In larger channels,
it is more frequent to encounter systems exhibiting external superfluid critical
velocities that have an inverse dependence on the channel dimension. These
follow, albeit loosely, the prediction of the Feynman model (solid line in fig-
ure 6.11) where a continuous shedding of energy is possible through vortices
positioned across the (micro)pores.
The linear dependence on temperature and the large critical velocities
found in our work are both consistent with a critical velocity of the inter-
nal type. Its qualitative behavior is also very similar to the extensive data
of Zimmermann et al. [100]. The functional dependence of the critical ve-
locity closer to zero temperature was not accessible in this work; the values
reported on figure 6.11 would be even higher if the extrapolated critical veloc-
ities at lower temperatures were taken as an estimate. Access to data closer
to T = 0 would have also allowed us to characterize the plateauing of vc below
T ' 150 mK, indicating perhaps the transition from a thermally activated
process to a quantum tunneling activation of the phase slip events [102, 103].
6.5 Summary
In summary, we have shown that measurements of liquid helium mass transport
in single nanopores can be interpreted with a two-fluid model containing a
132 Hydrodynamics of Superfluid Helium in a Single Nanohole
short-pipe viscous flow formalism. The extensive characterization of mass flow
in our nanopores can account for the non-negligible contribution of the normal
component of He-II. This has allowed us to infer the superfluid contribution
to the total flow, as well as its critical velocity. The latter was found to be the
largest ever observed in experiments of superfluid transport in channels.
133
Chapter 7
Conclusion
7.1 Summary
In this work, I have developed a novel measurement scheme for the charac-
terization of mass flow in single nanopores. The sensitive detection of the
mass flow was made possible by employing a mass spectrometer to count the
atoms passing through a very thin silicon nitride membrane in which a single
nanopore was drilled. The choice of a solid-state membrane was made to lever-
age technical innovations in the use of transmission electron microscopes that
allow for the controlled ablation of material with their high-intensity electron
beam. This technique makes it possible to fabricate nanopores of any desired
dimension through thin solid-state membranes.
By designing a transport experiment where two reservoirs are separated by
the wafer containing the thin membrane, very accurate measurements of mass
transport between the two reservoirs were made possible. Several experimental
cell designs were implemented in order to optimize the yield of the measure-
134 Conclusion
ment scheme and ensure that the wafer could remain completely sealed over
a broad range of temperatures and pressure. I have found that the most ef-
ficient technique of sealing the wafers to the metal body of the experimental
cell was to use small indium o-rings. This method was proven effective even
after several thermal cycles between room temperature and typical cryostat
temperatures.
The controlled introduction of helium in the source reservoir of the exper-
imental cell allowed us to control the pressure gradient across the nanopore.
The drain side of the experimental cell was continuously pumped such that any
helium passing through a nanopore could be detected by the mass spectrome-
ter. The combination of vacuum below and positive pressure above the mem-
brane induced mass transport determined by the conductance of the nanopore.
By measuring the mass flow and calculating that conductance under different
thermodynamic conditions, the physics of confined fluids was studied.
The first experiments performed were designed to test whether the mass
flow could be quantitatively understood. A large pore of ∼100 nm in diameter
was exposed to a large range of pressure differentials and the conductance of the
pore was monitored under these conditions. The conductance at low pressures
was found to be very accurately modeled by Knudsen effusion in a short-pipe.
A fit to the conductance data in this flow regime yielded an effective flow
area with radius within 1% of the dimensions found using a TEM image. The
agreement between the electronic imaging and the mass flow determination of
the radius was sufficiently accurate to confirm the validity of the measurement
scheme.
7.1 Summary 135
We further investigated the conductance of this sample at higher densities
where the flow characteristics were expected to be understood with the theory
of viscous transport in continuous media. From the conductance data over the
whole transition region, we were able to develop a phenomenological model
tailored for short-pipe channels where the effects of the geometry of solid-state
nanopores could not be neglected. We then extended our investigation to pores
well within the nanoscopic range. We presented results of direct measurement
of the gas transport in the nanopores as small as 15 nm. This is to our
knowledge the smallest dimension where such measurements were performed.
These experiments demonstrated the need to correct the Clausing factor to
account for the larger transmission probability caused by the tapered opening
of the nanopores. The very accurate monitoring of the nanopore dimensions
also allowed us to discover a slow structural deformation of the nanopores.
Some of the results from this tangential line of investigation are presented in
the appendix of this thesis.
Building on the results obtained with gas flow in single nanopores, we de-
signed a second experimental cell such that measurements could be realized
inside a 3He cryostat, at low temperature. Our results indicate that the liquid
helium flow through a nanopore is detectable with this technique. Further-
more, and as expected, we observed the mass transport below Tλ ≈ 2.17 K to
be much larger than in the viscous, normal flow of helium fluid. Using a two
fluid model of He II, we interpreted the additional mass flow as originating
from a superfluid component flowing with negligible viscosity. The effective
flow of superfluid through the nanopore provided an average velocity that we
136 Conclusion
collected at several temperatures between ∼1.55 K and Tλ, for both a 45 nm
and 15 nm diameter samples. These velocities were assumed to correspond to
the critical velocity, i.e. the velocity defined by the onset of dissipation mech-
anisms inside the superfluid. Our results do show a clear linear dependence on
temperature, and no significant difference between the two samples was con-
sistently observed. These inferred critical velocities are the largest recorded
to date in channel experiments. Our experiments of superflow in single chan-
nels have also extended the range of pore sizes available by nearly an order of
magnitude.
7.2 Future Work: Towards the Luttinger Regime
The experimental scheme developed in this work has been shown to be very
accurate for the determination of mass flow in nanopores as small as ∼15 nm.
In fact, a scaling of the signal-to-noise ratio with current flow level indicates
the technique can be extended to nanopores with a diameter as small as 1
nm. A quantitative determination of mass flow in the nanoscopic range offers
opportunities for other fields of study. For example, experiments with bio-
molecules such as DNA translocated through nanopores [104] are very sensitive
to the dimensions of the pores. In the transition from the microscopic world to
the nanoscopic one, some groups have observed anomalous transport [105, 106,
107, 108] and surprising effects [109] suggesting that the physical hypothesis
at the pore boundary conditions might need to be carefully considered.
Our ability to detect the signal from a single nanopore and the high pre-
cision offered by TEM drilling, are features of great interest for experiments
7.2 Future Work: Towards the Luttinger Regime 137
aimed at investigating corrections to hydrodynamics models. Currently, such
experiments are realized in carbon nanotubes and recent, surprising results
[110] will benefit from the availability of tailored channels like those presented
in this thesis.
Apart from the impact to classical fluid transport studies, pushing the
limit towards even smaller nanopores is greatly relevant for the ultimate goal
of implementing the long sought-after Tomonaga-Luttinger liquid with neutral
particles. With this in mind, future work will aim at mapping the character-
istics of helium mass flow in nanopores of various sizes, all the way down to 1
nm. Preliminary results from our group with nanopores as small as 6 nm show
how this work helps to bridge the gap towards the 1D limit; these pores are
within a factor of ∼2 of the dimensions required to observe Luttinger liquid
behavior, according to the quantum Monte Carlo simulations. In parallel to
this objective, the low temperature experiments can in principle be reproduced
using 3He, either as a pure fermionic fluid or as a diluted medium (a 4He-3He
mixture). These mixture offer interesting opportunities [111] that could be
directly supplemented by our ability to work with nanometric apertures and
could, in principle, demonstrate the quenching of quantum statistics in one-
dimension.
138
139
Appendix A
Appendix: Nanopore Shrinking
A.1 Blockages of Nanopores
As we endeavored to reproduce our experiments within smaller nanopores, we
regularly suffered blockage issues. Fabrication with the transmission electron
microscope (TEM) allows for an image to be taken within a few minutes of
the nanopore drilling. This helps to validate the dimensions of samples before
making mass transport experiments with them. On several occasions however,
the measurement of the mass flow through some samples would produce no
signal, except for an unwanted diffusion through epoxy with the earlier designs
of the experimental cell. Several attempts at thermal cycling theses blocked
samples or applying very large pressure gradients across the pores have never
been successful at re-opening those blocked pores. New images of these samples
with the TEM would subsequently confirm their blocked state.
140 Appendix: Nanopore Shrinking
A.1.1 Two Types of Nanopore Blockages
It is interesting to note that blocked pores can be easily detected with the
TEM imaging. This is in part because the silicon nitride membranes are
manufactured very flat and small variations of thickness produce a different
intensity of the electrons transmitted, which we observe as a varying saturation
of the pixels in the image. Indeed, the ablation of material from the membrane
in the area right around the nanopore makes the image appear lighter, and
the area further away from the pore becomes darker because of an unavoidable
accumulation of trapped charges on the surface of the membrane following the
exposure to high fluxes of electrons. When we observe a blocked nanopore
under the microscope, the position of the nanopores remains easily detectable
because of these specific patterns in pixel saturation. This variation in the
intensity of the image is easily observable by a trained eye and allows rapid
localization of the nanopore, even when the whole surface of the wafer is coated
with contaminants. Figure A.1 shows an example of a membrane coated with
a film of contaminants and in panel B, a close-up of the blocked nanopore.
As can be easily seen on this picture, contaminants cause a reduction of the
intensity of transmitted electrons. The dark spots on this picture are droplets
of a foreign substance.
Two distinct cases occurred when we imaged blocked nanopores. The
first case was typically observed in conjunction with the presence of hundreds
of droplets on the membrane surface (figure A.1A), usually several hundred
nanometers wide, and easily noticeable because of their peculiar reaction un-
der irradiation with the intense electron beam. Indeed, these droplets were
A.1 Blockages of Nanopores 141
visibly affected by the exposure to the energetic electrons, in contrast to the
immutability of solid substrates like the SiN and Si. A strong carbon signal
was observed when we performed an EDX analysis centered on nanopores with
this type of blockage, which led us to suspect a contamination from organic
molecules. These membranes could not be re-drilled because the contaminants
would typically flow back into the freshly drilled aperture, thus blocking the
nanopore instantaneously.
Fig. A.1 (A) Transmission electron microscope image of a free-standing membrane with droplets of contaminants. (B) Close-up ofthe nanopore coated with a thin film of contaminants. The irregulardiffraction pattern over the whole surface of the image indicates thenanopore is not open.
The second type of blockage was characterized by a very different appear-
ance of the SiN membrane. The TEM image would be almost identical to a
freshly drilled nanopore except that, instead of a uniform pixel saturation, the
area where the nanopore used to be would now display the typical diffraction
pattern of amorphous materials. This led us to conclude that a thin layer
of amorphous material blocked the aperture while the rest of the surface re-
142 Appendix: Nanopore Shrinking
mained unchanged. In order to compare to the first type of blockage, we again
performed analysis of the element composition in and around the nanopore,
and no organic compounds were detected, whereas large silicon and nitrogen
signatures dominated the signal as expected.
Contamination Prevention
The presence of organic material on the surface of the membrane was an issue
that we tackled by first designing filters and traps such that any gas sent to the
experimental cell from the gas handling system (GHS) would be filtered. These
filters were described earlier in chapter 4, and one such filter was positioned at
every entry point of the GHS. The GHS was then flushed1 several times, until a
final filling was done with ultra-high-purity (99.999%) helium gas. This clean
gas was then kept in the GHS and all operations were subsequently completed
in a closed-cycle manner by using the pumping capacity of the dipstick.
The discovery of the contamination of samples also led to a new experi-
mental cell design where no epoxies were needed and all joints were exclusively
made with compressed metal rings. I found the most effective design for the
experimental cell had all joints made with a compressed indium wire. These
additional precautions and experimental features led to the resolution of the
contamination issues.
1A process where a volume is emptied and partially refilled in order to remove foreignparticles. This process is usually repeated several times.
A.2 Partial Nanopore Shrinking 143
A.2 Partial Nanopore Shrinking
As was noted earlier in chapter 5, we noticed that on occasions, the conduc-
tance of the nanopores studied would decrease between two similar measure-
ments taken on two different days. In fact, the modeling of the conductance in
the Knudsen regime allowed us to detect a significant decrease in the size of the
nanopore over time. This decrease in diameter was, however not necessarily
leading to full blockage of the apertures. As was demonstrated by the multiple
samples with which we managed to observe mass flows, some nanopores could
be sufficiently stable to remain open for several months without any detectable
changes to their dimensions.
These results prompted us to further investigate the parameters leading to
stable nanopores. We used TEM imaging of nanopores as a means to measure
the evolution of the dimensions for samples of different thickness, from various
manufacturers and in different storage conditions. Specifically, we imaged
nanopores ranging in diameter from 5 to 50 nm over several days and weeks.
Figure A.2 shows a typical time-dependent structural relaxation in nanopores
with initial diameters of (A) 25 nm and (B) 12 nm. The observed decrease in
diameter was of 35% in (A), and in (B), I show a case where the blockage was
effectively complete. Both of these samples were fabricated in a 50 nm thick
SiN membrane under similar conditions. The TEM imaging shows a clear
decrease in the nanopore diameter over a time scale of the order of O(102)
hours.
As explained in the section on sample fabrication, it is a common practice to
expose nanopores to a mid-range intensity of electron bombardment to induce
144 Appendix: Nanopore Shrinking
0 5004003002001000
20
40
60
Time t [hours]
Dia
met
er d
[nm
]
0 h 48 h 96 h 168 hA
B
C
Fig. A.2 TEM images of a nanopore as a function of time. Thenanopore diameter at fabrication was 25 nm in (A) and 12 nm in(B). The nanopores were drilled in 50 nm thick amorphous siliconnitride membranes and stored at 77 K between each imaging ses-sions. All scale bars are 10 nm long. (C):Diameter of several suchnanopores as a function of time after fabrication. Open symbolsare for samples stored at liquid nitrogen temperature and the filledsymbols are for samples kept at room temperature. The symbolsat zero diameter indicate nanopores that are completely blocked.The lines connecting the symbols are guides to the eye.
fluidization of the material around the pore and reshape them in the process.
This is part of a normal drilling procedure. There is a legitimate question
regarding the risk of affecting nanopores during imaging with the TEM. The
decrease in diameter observed in figure A.2A,B could however not have been
A.2 Partial Nanopore Shrinking 145
caused by the exposure during imaging because the intensity of the electron
beam required to image the pores was kept approximately 10 times lower than
that required to reshape the surface. We specifically tested this hypothesis by
exposing a 9 nm nanopore, for a total period of 40 minutes, to a beam intensity
equivalent to that used during regular imaging of nanopores. No noticeable
change in the dimensions of the pore were observed. The decrease in diameter
observed in figure A.2 was thus caused by a slow structural relaxation of the
amorphous silicon nitride independent of our measurements.
Figure A.2C shows the diameter of nanopores as a function of the time
elapsed since their fabrication, for two distinct storage conditions. Specifically,
we stored some wafers under vacuum at liquid nitrogen temperatures (open
symbols) and others remained at room temperature (filled symbols). The
samples at low temperature were only warmed up for the imaging sessions.
All the nanopores shown in figure A.2C were fabricated in a 50 nm membrane,
and the storage temperature is found to have a negative correlation with the
rate of pore size decrease.
Two types of behavior are identifiable in the structural relaxation of the
nanopores. The relaxation can either lead to a stable nanopore or a blocked
one. The two distinct behaviors are easily distinguished in figure A.2C where
the four higher datasets correspond to stable nanopores and the bottom ones
show blocked pores. In order to characterize these two types of behavior, we
fabricated several nanopores in membranes of different thickness between 10-
100 nm. Figure A.3 shows the diameter of nanopores drilled originally with
diameters between 25 and 50 nm and their time-dependence. These nanopores
146 Appendix: Nanopore Shrinking
were all opened, even after several hundred hours. The relaxation of these
nanopores approaches an equilibrium point, and the dashed lines are fit to the
data for a phenomenological equation with a simple exponential decay
d(t) = df + (d0 − df )(τt
)(1− e−
tτ
),
where df is the final diameter, d0 the initial diameter and τ a characteristic
time. This equation can be used in the instances where the initial diameter
is large enough that the nanopores approache a near steady-state equilibrium
shape.
Fig. A.3 Diameter of nanopores as a function of the durationsince fabrication. These nanopores asymptotically approach a nearsteady-state through a structural relaxation. The dashed lines arefit with an equation with a single exponential decay. The insetshows the final diameter as a function of the initial diameter im-mediately after drilling and linear fit through the data.
The characteristic time of relaxation was found to vary between ∼40-160
A.2 Partial Nanopore Shrinking 147
hours and this range was observed for pores of the same size drilled in mem-
branes from different manufacturers, which suggests the process of amorphous
silicon nitride deposition affects the structural stability of the nanopores. This
also indicates the relaxation is closely linked to the microstructure of the ma-
terial and how the electron beam exposure affected it during fabrication.
In practical terms, the final diameter of stable pores was found to depend
nearly linearly on the initial pore size. The inset of figure A.3 shows the final
diameter as a function of the initial one for several pores that reached an
equilibrium shape.
30
20
10
00 100 200 300 400
Dia
met
er d
[nm
]
Time t [hours]
Fig. A.4 Diameter of nanopores as a function of the elapsed timefrom fabrication. The open symbols represent samples stored at lowtemperature, whereas the filled ones are for sample kept at roomtemperature.
As seen above in figure A.2C, the three smallest nanopores relaxed to a
fully occluded state within the first 100 to 200 hours after fabrication. In
these cases, the diameter has a distinctly different functional dependence on
time. Figure A.4 shows the size of several nanopores <25 nm as a function of
148 Appendix: Nanopore Shrinking
time, exclusively for samples that eventually closed. Again the open symbols
are for samples stored at 77 K and the filled symbols are for storage at room
temperature. The length of the error bar through the dots shows the uncer-
tainty on diameters measured from TEM images. The gradual and steady
decrease in diameter for these samples was proportionally faster than in the
case of larger pores.
A.3 Conditions for Stable Nanopore Fabrication
0 10 20 30 40 50 600
20
40
60
80
100
Leng
th L
[nm
]
Diameter d0 [nm]
Fig. A.5 Phase diagram of the structural stability of nanoporesas a function of the membrane thickness. The open symbols corre-spond to nanopores that remained open and the filled symbols cor-respond to nanopores that eventually became completely blocked.
From these results it became evident that the thickness of the membrane
and the diameter of the nanopore could be used to determine a “boundary”
across which the type of relaxation changed from a slow approach to an equi-
librium, to a relatively fast shrinking of the aperture. We mapped the state
A.3 Conditions for Stable Nanopore Fabrication 149
of several samples after hundreds of hours, and for several membrane thick-
nesses. Figure A.5 shows this phase diagram, where the open circles correspond
to nanopores that remained open, and the filled ones to blocked nanopores.
This diagram shows the difficulty of keeping small nanopores open for ex-
tended periods of time. While we did not have many samples with thicknesses
below 30 nm, we note that nanopores with initial dimensions between 15-20
nm are hard to keep open for extended periods of time. A corollary of this is
that in order to make experiments with nanopores barely a few nanometers
across, it is recommended that thinner membranes be used as substrates.
150
151
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